The symplectic hook-content product conjecture

Let $\lambda$ range over partitions, let $h^{\lambda}(u)$ denote the hook length of a cell $u\in\lambda$, and let $c_{sp}^{\lambda}(u)$ be its symplectic content. For an indeterminate $t$, define $\binom{t}{2}=t(t-1)/2$ and $\binom{t+1}{2}=t(t+1)/2$. Symplectic hook-content product conjecture. For $t$ an indeterminate, one has

$$\sum_{n\geq0}q^n\sum_{\lambda\vdash n}\prod_{u\in\lambda}\frac{t+c_{sp}^{\lambda}(u)}{h^{\lambda}(u)}= \prod_{j\geq1}\frac{(1-q^{8j})^{\binom{t+1}2}}{(1-q^{8j-2})^{\binom{t+1}2-1}} \left(\frac{1-q^{4j-1}}{1-q^{4j-3}}\right)^t \left(\frac{1-q^{8j-4}}{1-q^{8j-6}}\right)^{\binom{t}2-1}.$$

The article proves the case $t=0$, while the identity for general indeterminate $t$ remains conjectural.