Does coordinate acceleration match proper acceleration when $v=0, s=0$?

Question

Alice and Bob are riding in their rocket at arbitrary proper acceleration through an arbitrary region of spacetime.

Bob steps out of the space ship right next to Alice, such that at $t=t'=0$, $v=0$, displacement $s=s'=0$ and Bob's proper acceleration is $0$. Alice's proper acceleration is $A$.

Alice gets the primed frame.

At that moment, do Bob and Alice measure Alice and Bob respectively accelerating away with coordinate acceleration $a(t=0)=a'(t'=0)=A$ for all $A$ in all regions?

Context: This question came to mind in relation to the comments on my old answer here.

Answer 1

Yes, that is the definition of proper acceleration which is the coordinate acceleration with respect to the instantaneously comoving inertial frame. In this frame, the proper acceleration has temporal component of zero and spatial components equal to the coordinate 3-acceleration. So Bob is in the instantaneously comoving inertial frame of Alice. So, in an infinitesimal time interval at that instant, they observe each other to have equal and opposite accelerations.

Answer 2

Yes. We can think for example that Alice and Bob are both in a airplane. In the moment that Bob jumps, Alice has a coordinate acceleration $g$ upward for him. Bob has a coordinate acceleration $g$ downward for her.