And if they are Christian, it will probably be Nestorian or some such - at any rate, something viewed as heresy by Constantinople.
As for whether Persia "recovers" or not, that hardly matters. If it doesn't, it will only be replaced by something else. There'll always be a power there.
That's how it always went. The Romans crushed the Seleukids and got the Parthians in their place. They outlasted the Parthians only to get the Sassanids in their place. They beat off the Sassanids only to get the Arabs in their place, when the Arabs declined they got the Seljuks in their place and when the Seljuks declined they got the Ottomans in their place. There was always another opponent waiting just behind the current one.
That’s the nature of geopolitics though... Every major nation that competes for hegemony will encounter this issue. Nations who are either:
1. Isolated to a large degree, at least from powerful political foes. Examples such as the Inca Empire faced very little stare actor opponents to authority, primarily due to the relative isolation it enjoyed and technological/societal superiority to nearby tribes. Some other examples would be the Edo period Tokugawa Shogunate.
2. A state that has reached a certain level of hegemony that predominantly, only unlikely occurences cause collapses or internal disasters. Examples of these are the zenith of Rome and the Achaemenid Empire. Both of which were virtually untouchable for their time, in terms of total defeat in war unless the occurrence was very irregular. The Mongol Empire is one such state and conglomeration that in essence, lacked geopolitical foes who could truly destroy the Mongol ensemble completely, however the empire fractured due to internal contradictions.
3. The power is so shielded by another or inconsequential to amount to a non threat and thus accumulation of rival political entities, is limited. Examples of these are much of the Chinese spheres tributaries, especially the various Tibetan states. San Marino and Switzerland would be some other examples, both lack political aspirations that would warrant rival accumulation.
In regards to Sassanid v. Byzantine geopolitics, the issue depends.
Yes, the Sassanid empire has less borders with varying groups than Byzantium. However, this does not mean the Sassanid threats or power is greater long term or short term. To make one statement, Byzantium is simply stronger and more robust than the Sassanid. This is a simple reality and thus all discussions must build from the knowledge that the pain Byzantium can take is possibly too much for the Sassanid and that in protracted wars and conflict, Byzantium is likely to be the winner in such scenarios.
Sassanid threats analyzed:
-Rival clans revolt. Self explanatory, this is worse in the lands of Parthia, in terms of threat posed.
-Byzantium itself...
-Invasion from the Caucasian mountains. This, I mean, an invasion from the steppe bypassing the mountains and striking into Iran or Iraq. This happened numerous times, likely the Hittites, Cimmerians, Scythians, Khazars, etc... Sassanid protection of the frontier depends on maintaining various fortresses and chokepoints. If Sassanid order collapses, without ordered restructuring, the route is open for the north.
-Eastern: A myriad of foes and possibly ones. Migrating Turkic hordes, Mongol hordes, the rising Tibetan Empire, the Zunbil, the fading yet remaining Hepthalite Empire, the Gupta and other Hindu-Buddhist powers originating from beyond the Khyber.
-Southern: The threat that broke the Sassanid state in otl. This would be the possibility of Arabic conquest.