Headsup Vs. Nutjob - postflop strategy UI

Later I'm going to post some preflop hands vs. a guy who raises/caps EVERYthing preflop, but here, suppose you've decided to go in with Ace-high or a micro-pair (55-22).

The nutjob NEVER folds - you cannot bluff or semi-bluff. You bet or call strictly for value.

[He raises/caps anything on the first two streets, and bets out turn/river unless you cap the flop. In those cases, he may calldown with anything, and raises with marginal holdings (midpair, etc.)]

Now, if your Ace-high or micro pair is unimproved, does it make sense to call down to the river all the time? Some of the time? With what frequency?

Obviously, if there's significant improvement (you hit mid pair or better) you may bet/raise/3-bet/cap...but what about unimproved ace-high and micro-pairs? Check/call all the way?

How about unimproved king, queen or jack high?