Does a ( min,+)-WA preserve REG by inverse image?

We are interested in functions realized by weighted automata over the semiring $\mathbb N_{\mathsf{ min}}=\langle \mathbb N\cup\left\{ \infty \right\} , \mathsf{ min}, +, \infty,0\rangle$. An $\mathbb N \mathsf{ min}$-weighted automaton $\mathcal A$ over alphabet $\Sigma$ realizes a partial function $[\![ \mathcal A ]\!]:\Sigma^*\rightarrow \mathbb N$. We want to decide given such a function if it preserves ``simple'' sets by inverse image. By simple we mean regular languages ($\mathbb N$ is viewed a the set of words over a unary alphabet, hence the regular languages are the semilinear sets). Let us state the decision problem. Input: $\mathcal{A}$, weighted automaton over $\mathbb{N}_{\mathsf{min}}$ Question: Does it hold that for all $S\subseteq \mathbb N$ semilinear, $[\![ \mathcal A ]\!]^{-1}(S)$ is regular? Remark. This is equivalent to solving the problem over the semiring $\mathbb N_{\mathsf{ max}}=\langle \mathbb N\cup\left\{ -\infty \right\} , \mathsf{ max}, +, -\infty,0\rangle$.