Valuation formula for Kac polynomials of multi-arrowed quivers

Let $Q=(I,d4aa)$ be a quiver with imaginary vertices $I^{\operatorname{im}}$ and real vertices $I^{\operatorname{re}}$, let $\underline{n}\in(\operatorname{\mathbf{N}}_{\geq 1})^{\Omega}$, and let $Q_{\underline{n}}$ be obtained by replacing each arrow $\alpha:i\to j$ by $n_\alpha$ arrows. For a dimension vector $\operatorname{\mathbf{d}}=(d_i)_{i\in I}$, let $A_{Q_{\underline{n}},\operatorname{\mathbf{d}}}(q)$ be the associated Kac polynomial, and define the valuation of a nonzero polynomial as its smallest exponent with nonzero coefficient.

Valuation conjecture. Provided $A_{Q_{\underline{n}},\operatorname{\mathbf{d}}}(q)\neq 0$, its valuation is

$$v_{\underline{n}}=\sum_{i\in I^{\operatorname{im}}}\left(1+d_i\left(\sum_{\alpha:i\to i}n_\alpha-1\right)\right).$$

This conjecture is supported by computations for multiloop and tennis-racket quivers. It concerns the lowest power of $q$ occurring in Kac polynomials as the numbers of parallel arrows vary.