Decidability of ( ,+)-weighted automata determinization

Is the following problem decidable: Given $A$, a weighted automaton (WA) over the $(\min,+)$ semiring, answer whether $A$ has an equivalent deterministic WA? Remarks: The problem is equivalent when considering only automata with weights in $\{0,1\}$ (a.k.a. distance automata). The problem is known to be decidable for polynomially-ambiguous WA. For the general case of exponentially-ambiguous WA, we don't even have any interesting examples where determinization incurs a significant state explosion.