It was suspected as early as the 1360s (within a decade of the document coming to light, basically) that the Hapsburg Privilegium Majus was a forgery. However, there are two caveats here. One is that the Privilegium Majus isn't a single document, but five distinct documents, claimed to date from the 10th or 11th centuries. The other factor here is that at the time it was "discovered" in 1359, the Privilegium Majus was not an assertion of novel rights and privledges, but as instead intended to legitimize what was already the de-facto political powers of the Hapsburg dynasty within the political structure of the Holy Roman Empire

Some context may be helpful here. In 1273, a fairly minor imperial count, Rudolf IV of Hapsburg, gets elected "King of Germany" and becomes Rudolf I of Germany. Rudolf's godfather was Emperor Fredrick II, of the House of Hohenstaufen. When Frederick II died in 1250, the powerful Hohenstaufen dynasty dies with him, and this triggers a period of nearly 25 years of political instability in which there is no King of Germany, and no official Holy Roman Emperor. Suffice to say that Rudolf was a compromise candidate that helps end a period of political instability in Central Germany, which was mainly helped by the fact that he had several daughters which married into other prominent ducal houses. This marks the rise of the Hapsburgs from being a minor noble family centered mainly in what is today Switzerland and Swabia to a major ducal (and eventually royal) house centered mainly in what is today Austria and Hungary. Around 1280, Rudolf I of Germany uses his powers as King of Germany to make his two sons, Albert and Rudolf Jr, dukes of Austria and Styria. Rudolf Jr dies at the age of 20, but he has a son, John (aka Johannes of Swabia).

So then Rudolf Senior dies in 1291, having been only marginally successful in bringing peace to the lands of the Germanies. He get succeeded as King of Germany by Adolf of Nassau, whom gets deposed a few years later in 1298 by the same people that elected him, essentially for not being a sufficiently cooperative puppet of said powerful ducal electors. Albert of Hapsburg, Duke of Austria and Styria, gets elected King of Germany in 1298, and rules fairly successfully, trying to bring vacant fiefs back under imperial administration, and of course sucking up shamelessly to the pope. Albert I of Germany get assassinated in 1303 by his nephew, John of Swabia, hereafter known as John Parricide, because John thinks Albert I is trying to deprive John of his rightful inheritance of Styria and Bohemia.

The assassination of Albert I of Germany triggers another set of political headaches, and when the dust settles in 1312, Henry VII of the house of Luxembourg as been elected both Holy Roman Emperor and King of Germany. For the next two hundred years the title of Holy Roman Emperor switches back and forth between the dynasties of Luxembourg, Hapsburg, and Wittelsbach, as they vie for political dominance in Central Europe. Eventually the house of Hapsburg comes out on top, and after 1500 the office of Holy Roman Emperor increasingly becomes a Hapsburg patrimony, although the poltical process of electoral monarchy continues as a formality until the dissolution of the HRE in the early 19th century.

In this context, the Privilegium Majus is a political maneuver to assert the legitimacy and "inviolability" of continued Hapsburg rule over Bohemia, and most especially over Austria. So even if it was a forgery, it was one that merely legitimized what was already the de facto state of affairs, and contesting it would have created more political conflict than it was worth, from the perspective of the other competing dynasties at the time.