Well, the answer is no, and also yes, because we know very little for certain about Alexander as a person. This sounds weird, but let me explain. Because we have so many contradictory sources for Alexander, it is often difficult, if not impossible, for us to determine which specific telling of an event is closest to the truth. For instance, we have two completely different and irreconcilable versions of the Battle of the Granicus. And if we can't even get the details of a battle straight, then our biographical portraits of the man are going to be even more of a mess. In broad terms, the sources for the life and times of Alexander can be divided into two traditions: an 'official' tradition based on pro-Macedonian sources, and a 'Vulgate' tradition based on largely anti-Macedonian sources but with varying degrees of pro-Alexander gloss applied. The principal representatives of the former camp are Arrian of Nikomedia's Anabasis of Alexander and Plutarch's Life of Alexander, while the latter includes Book 17 of Diodoros of Sicily's Library of History, Quintus Curtius Rufus' History of Alexander the Great, and Justin's epitome of Books 10 and 11 of the Philippic Histories of Pompeius Trogus. Unsurprisingly, the soothsayer claim comes from the Vulgate.

But in order to put it in context, we need to look at all five versions of the event, with the relevant section being the lead-up to Alexander scaling the ladders. Note that not all of the sources actually give it in any significant detail.

On the following day Alexander divided the army in two for the attack on the wall, leading one division himself with Perdiccas in command of the other. At this first stage the Indians did not resist the Macedonian onslaught, but abandoned the outer defences and congregated for refuge in the citadel. Alexander and his men now broke open a small gate and got inside the city long before the other troops under Perdiccas, who were slowed by their difficulty in negotiating the wall, as most of them had not even brought ladders, in the belief that the city had already been taken when they saw the outer walls denuded of defenders. But when it became clear to Perdiccas’ men that the citadel was still in enemy hands, and they could see large numbers marshalled for its defence, they made every effort to force their way in – undermining the wall, setting up scaling-ladders wherever they could. Alexander thought the Macedonians bringing up the ladders were too slow about it, so he seized a ladder from one of them, set it up against the wall himself, and, huddled under his shield, climbed up: Peucestas came up after him bearing the sacred shield which Alexander had taken from the temple of Athena at Troy and kept always with him, having it carried before him in his battles. He was followed up the same ladder by Leonnatus the Bodyguard, and Abreas, one of the soldiers on double pay, mounted by another ladder.

– Arrian, Anabasis 6.9.1-3

However, in attacking the people called Malli, who are said to have been the most warlike of the Indians, he came within a little of being cut down. For after dispersing the inhabitants from the walls with missiles, he was the first to mount upon the wall by a scaling ladder, and since the ladder was broken to pieces and he was exposed to the missiles of the Barbarians who stood along the wall below, almost alone as he was, he crouched and threw himself into the midst of the enemy, and by good fortune alighted on his feet.

– Plutarch, Life of Alexander 63.2-3

Alexander neared the first city and thought to take it by storm, but one of the seers, named Demophon,​ came to him and reported that there had been revealed to him by numerous portents a great danger which would come to the king from a wound in the course of the operation. He begged Alexander to leave that city alone for the present and to turn his mind to other activities. The king scolded him for dampening the enthusiasm of the soldiers, and then, disposing his army for the attack, led the way in person to the city, eager to reduce it by force. The engines of war were slow to come up, but he broke open a postern gate and was the first to burst into the city.​ He struck down many defenders and, driving the others before him, pursued them to the citadel.

The Macedonians were still busy fighting along the wall. Alexander seized a ladder, leaned it against the walls of the citadel, and clambered up holding a light shield above his head. So quick was he to act that he reached the top of the wall before the defenders could forestall him. The Indians did not dare to come within his reach, but flung javelins and shot arrows at him from a distance. He was staggering under the weight of their blows when the Macedonians raised two ladders and swarmed up in a mass, but both broke and the soldiers tumbled back upon the ground.

– Diodoros 17.91.2-6

After this the Macedonians came to the capital town of the Sudracae. Most of the enemy had sought refuge here, though their confidence in its walls was no greater than their confidence in their arms. Alexander was already making his move towards the town when a seer began to issue warnings against the siege which, he said, the king should at least postpone since it was predicted that his life was in danger. Alexander looked at Demophon (that was the seer’s name). If someone interrupted you like this,’ said the king, ‘when you were preoccupied with your craft and observing the entrails, I am sure you would consider him an exasperating nuisance.’ After Demophon replied that such would certainly be the case, Alexander continued: ‘When I have my mind on weighty matters and not on animal intestines, do you think anything could be a greater hindrance to me than a superstitious seer?’ Waiting only to give this reply, he ordered the ladders to be taken forward and, as the others hesitated, scaled the wall.

– Curtius 9.4.26-30

...next he sailed to the Mandri and Sigambri, who met him with eighty thousand foot and sixty thousand horse. Gaining the victory in a battle, he led his army against their city; and supposing, as he looked from the wall, which he had been the first to mount, that the place was destitute of defenders, he leaped down into the area of the city without a single attendant.

– Justin 12.9.3-5