The symplectic and orthogonal squared-content identity conjecture

Let $\lambda$ range over partitions, let $h^{\lambda}(u)$ denote the hook length of a cell $u\in\lambda$, and let $c_{sp}^{\lambda}(u)$ and $c_{O}^{\lambda}(u)$ denote its symplectic and orthogonal contents, respectively. Symplectic and orthogonal squared-content identity conjecture. For an indeterminate $t$, one has

$$\sum_{n\geq0}q^n\sum_{\lambda\vdash n}\prod_{u\in\lambda}\frac{t+(c_{sp}^{\lambda}(u))^2}{(h^{\lambda}(u))^2}= \prod_{j\geq1}\frac1{(1-q^{4j-2})(1-q^j)^t}= \sum_{n\geq0}q^n\sum_{\lambda\vdash n}\prod_{u\in\lambda}\frac{t+(c_{O}^{\lambda}(u))^2}{(h^{\lambda}(u))^2}.$$

This is presented as the symplectic and orthogonal counterpart to Stanley's hook-content identity and remains open in the source.