Conjectured convolution inverse for the Tutte q-polynomial transform

Let $R$ be the rectangular poset formed by the product of the chains $0,\dots,\rho$ and $0,\dots,\nu$. For intervals $[a,b]$ and $[c,d]$ in $R$, let $\alpha$ be the convolution kernel determined by

$$\alpha(a,b;c,d)={\binom{a}{c}}_{q}{\binom{b}{d}}_{q}q^{(a-c)(b-d)}.$$

The convolution inverse $\beta$ is defined by $\alpha\circ\beta=\beta\circ\alpha=\delta$, where the identity kernel is $\delta_a^c\delta_b^d$.

Conjectured convolution-inverse formula. For the relevant indices $a,b,c,d$,

$$\beta(a,b;c,d)=(-1)^{(a-c)+(b-d)}{\binom{a}{c}}_{q}{\binom{b}{d}}_{q}q^{\binom{|(a-c)-(b-d)|}{2}}\left(1+q^{|(a-c)-(b-d)|}-q^{\max(a-c,b-d)}\right).$$

This formula would invert the rank-generating-function transform relating the Tutte $q$-polynomial to the rank generating function of a $q$-matroid. It was obtained from computational experimentation and is presented as a conjecture; no proof or resolution is supplied in the source.