Support caching for branch names in new PR view

[ulugbekna]

It would be great if the target branch picker could show branches faster, especially on subsequent opens.

It's hard to pin point how caching should work but I would expect to see the main and base of current branch to be visible immediately.

See:

CleanShot.2025-12-17.at.12.15.04.mp4

GHPR 0.125.2025121704