The balanced-partition conjecture for the $\mathsf a;q$-down-degree expectation

Let $\mathsf a$ and $q$ be indeterminates. For a balanced partition $\lambda$, write $w=\lambda_1$ for its width, $\ell=\ell(\lambda)$ for its length, and $d=\gcd(w,\ell)$. Define

$$W_{\mathsf a;q}(n)=\frac{1-\mathsf a q^{1+2n}}{1-\mathsf a q}q^{-n},\qquad [n]_{\mathsf a;q}=\frac{(1-q^n)(1-\mathsf a q^n)}{(1-q)(1-\mathsf a q)}q^{1-n}.$$

For $x,y\in[\emptyset,\lambda]$ with $y\lessdot x$, if $y$ is obtained by deleting a cell in row $s$, set

$$\operatorname{wt}(x,y)=W_{\mathsf a q^{\frac{w+\ell}{d}};q}\left(\frac{w(s-1)+\ell(|x|-1)}{d}\right).$$

Define

$$\operatorname{ddeg}_{\mathsf a;q}(x)=\sum_{y\lessdot x}\operatorname{wt}(x,y),$$

with the sum over $y\in[\emptyset,\lambda]$, and

$$R(\lambda\mid\mathsf a;q)=\sum_{x\in[\emptyset,\lambda]}W_{\mathsf a q^{\frac{w\ell}{d}};q}\left(\frac{\ell|x|}{d}\right),$$

so that

$$\mathbb E_{\mathsf a;q}(X)=\frac{\sum_{x\in[\emptyset,\lambda]}\operatorname{ddeg}_{\mathsf a;q}(x)}{R(\lambda\mid\mathsf a;q)}.$$

Balanced-partition product conjecture. If $\lambda$ is balanced, then

$$\mathbb E_{\mathsf a;q}(X)=\frac{\left[\frac{w\ell}{d}\right]_{\mathsf a q^{\frac{w+\ell}{d}};q}}{\left[\frac{w+\ell}{d}\right]_{\mathsf a q^{\frac{w\ell}{d}};q}}.$$

This is an $\mathsf a;q$-analogue of the down-degree expectation formula. The source presents it as a conjecture for balanced partitions of arbitrary slope; no resolution is supplied.