Exponential positivity conjecture for nonnegative matrices

Let $C$ be an $r\times r$ matrix with non-negative entries, define $\boldsymbol{\alpha}!=\prod_{i=1}^r\alpha_i!$ for $\boldsymbol{\alpha}\in\mathbb{N}^r$, and let $\underline{T}$ act by $\underline{T}(x^{\boldsymbol{\alpha}})=q^{\boldsymbol{\alpha}^{t}C\boldsymbol{\alpha}}x^{\boldsymbol{\alpha}}$. Suppose that $a_{\boldsymbol{\alpha}}(q)\in\mathbb{N}[q]$ for $\boldsymbol{\alpha}\in\mathbb{N}^r$ and that

$$\exp\left(\frac{\sum b_{\boldsymbol{\alpha}}(q)\frac{x^{\boldsymbol{\alpha}}}{\boldsymbol{\alpha}!}}{q-1}\right)=\underline{T}\exp\left(\frac{\sum a_{\boldsymbol{\alpha}}(q)\frac{x^{\boldsymbol{\alpha}}}{\boldsymbol{\alpha}!}}{q-1}\right).$$

Exponential positivity conjecture. Then $b_{\boldsymbol{\alpha}}(q)\in\mathbb{N}[q]$ for every $\boldsymbol{\alpha}\in\mathbb{N}^r$. The conjecture was proved by Markus Reineke for $r=1$ and remains open for general $r$.