Alliance vs. Engines Pt.2 endgame analysis (move 45)

Black's next move is 44...Rc3+.  Note that this is a correction from an early post that said it was 44...Rxh3 directly.  (The engine plans to take on h3, apparently, after our king has moved away.)

Here's the present situation on the board (and this time the board is not static, so you can use it as an analysis board too):

[EDIT: This post refers to an earlier version in which we were incorrectly told black's move; it's not 44...Rxh3 directly, as my message below assumes.]

In previous analyses we had looked at 45.Rb1+ with the idea to activate our rook and penetrate into black's 7th rank from where we can both attack their king-side pawns as well as bother their king with checks from the back.

I wonder if 45.Re1 would be a useful alternative here -- obviously threatening black's e4-pawn (which they can only defend with ...Rc3+ followed by ...Rc4), while freeing space for the white king to take guard on b1 or a2 in the near future.

Thoughts?

A late correction, with apologies from Jorg about incorrectly transmitting the message at first.  He had neglected to mention that the engine's plan was to first give us a check 44...Rc3+, before taking the h3-pawn.

So the actual engine's move now is 44...Rc3+.  I've reflected this now on the initial diagram above.

there are only 3 legal moves here; of them:

Kb1 kills our Rb1+ plan

Kd1 and Kd2 allow it; after ...Rxh3, Rb1+ however Black has available ...Rb3 it hasn't until our K was in c2 and should we play Kc2 (or Kc1) after ...Rxb1, Kxb1 ...Kb3 I'm afraid we have to work out another plan for our R

urvang:  The end of your proposed line is mangled (starting from move 48).  What did you mean there? 

(Obviously, white's 48.Re1 would be met with Rxf2+ and black should win easily, but also white cannot legally move 48.Kb1 because our king is on d2, not next to b1.)

urvang:  There is also 47...Rb2+ for black, to be followed by ...Kb3 and then pushing their a-pawn forward.

In any case, for now I agree that we need to move 45.Kd2, in order to leave the 1st rank as the potential moving path for our rook.

Let's all who participate try to agree on our next move (we seem to have an early agreement) within the next 12-24 hours, so I can communicate it to Jorg.

for the time being let's confirm Kd2