Is it decidable whether the langauge accepted by a reversal-bounded counter machine is deterministic? I'm wondering if anyone can point me to either an algorithm or an undecidability proof for the following question:

Given a non-deterministic reversal-bounded multicounter machine $M$,
is there some deterministic reversal-bounded multicounter machine $M'$
such that $L(M) = L(M')$?

For anyone interested, reversal-bounded counter machines are described at length
in Oscar Ibarra's Paper about them.
I found a similar argument for context-free languages, which can be used here.
It is not decidable, since universality for non-deterministic reversal-bounded multicounter machines is undecidable, but becomes decidable for the deterministic version.

Suppose we can decide if $M$ can be determinized, and perform the determinization if it can.
Then we can decide if $L(M) = \Sigma^*$ as follows:

If $M$ cannot be determinized, then clearly it is not universal, since there is a deterministic machine accepting $\Sigma^*$.
If $M$ can be determinized, then we determinize it and test if it is universal.

This question on CS.SE give the analogous argument for CFL vs. DCFL.