It seems to me to be entirely incorrect. I can only talk about the sixth and seventh centuries, but war most definitely was not a constant, even in these two war-torn centuries. The first phase, the Anastasian War of 502-6 (named after the Roman emperor Anastasius), was the first large-scale conflict between the two empires after only two minor conflicts in the fifth century, so by then peace was the norm, not the exception. The interbellum period after 506 was admittedly quite tense, with imperial proxies engaging in combat from c.525 onwards, with open conflict beginning in 530. This war then ended in 532 with an 'Eternal Peace'.

This peace did not last very long, since the Persians launched a surprise attack in 540. This war dragged on until a peace was made in 562. Emperor Justin II however was keen on fighting the Persians, igniting a new conflict in 572, which lasted until 591. This may well have led to a long period of peace between the two empires, since Emperor Maurice had essentially helped Khusro II back onto his throne after the young Persian ruler was overthrown by an usurper, but Maurice's execution in 602, at the orders of the Roman usurper Phocas, extinguished all hopes of that. In 603 the 'last great of late antiquity' began, as Khusro used this opportunity to 'avenge' his benefactor, a conflict that consumed the eastern Mediterranean world and which finally ended in 628.

These late antique wars were certainly long and altogether we can only find a few decades of the two empires not fighting each other, but they were clearly different conflicts fought for different reasons by different people. Any attempt to conflate them together does not do them justice and creates the impression of two empires unable to make peace, when in reality co-operation was the norm in the century prior to the Anastasian War and even in these two centuries, during times of peace the two empires were quite content to help each other for their mutual benefit. They were rivals yes, but they also recognised that the two great empires were the 'two eyes of the world', that their opponent was the only power civilised enough to be treated as (near-)equals. There is simply no reason to view these two empires as two that were locked in conflict for centuries.