I assume so because that's how he would've aged. Stayed with her all those years up to 2024.
Well then...
Correct me if this is wrong:
1. 2024 Cap goes back to 1950.
2. At that time, OG Capsicle is frozen and stays frozen the entire time 2024 Cap is there living out his dream life with Peggy Carter. There are TWO Steve Rogers at this point; a frozen one and the one that time-traveled from 2024. Both would have been close enough in age that nobody would probably tell the difference, assuming he did pull off a switcheroo.
3. 2024 Cap leaves 1950's Peggy 70 years later to return to 2024 and sits on a bench, wrinkled and gray and frail, with a shield, which since it's 2024 when he left the branched timeline as well, he could have gotten from either Howard Stark, Tony Stark or Wakanda or maybe even just made a new one himself for giggles. He had 70 years, afterall, to do a lot of stuff.
--Makes you wonder - what was the fate of OG Capsicle? Was he ever found and rescued? 2024 Cap seems to have been there beyond the point that OG Capsicle was rescued in the current timeline. Does he prevent Capsicle from being found and just leaves him for dead the whole time while he lives out the life that he would have lived had he not been frozen??
There are of course a million other questions relating to Steve Rogers and the whole Marvel universe now, in that branched timeline. Like, do the Avengers even exist there? Did Steve Rogers continue to be Captain America Superhero when he was there all those years schmoozing peggy Carter? Does he fight Thanos AGAIN in that branch?
There is so much stuff to think about. But I'm mostly wondering what happens with OG Capsicle in the branched timeline.