Weighted average-size conjecture for self-conjugate core partitions

Let $s$ and $t$ be coprime positive integers. Write $\mathcal{D}_s$ for the set of self-conjugate $s$-cores, let $\mathcal{D}_t$ be the set of self-conjugate $t$-cores, and let $\operatorname{StabSC}_{s,t}(\lambda)$ be the stabiliser of $\lambda\in\mathcal{D}_s\cap\mathcal{D}_t$ under the level-$t$ action of $\widetilde{\mathfrak H}_s$ on $\mathcal{D}_s$.

Weighted self-conjugate core-average conjecture.

$$\frac{\displaystyle\sum_{\lambda\in\mathcal{D}_s\cap\mathcal{D}_t}\frac{|\lambda|}{|\operatorname{StabSC}_{s,t}(\lambda)|}}{\displaystyle\sum_{\lambda\in\mathcal{D}_s\cap\mathcal{D}_t}\frac{1}{|\operatorname{StabSC}_{s,t}(\lambda)|}}= \begin{cases} \dfrac{(s-1)(t^2-1)}{24} & \text{if } t \text{ is odd},\\[9pt] \dfrac{(s-1)(t^2+2)}{24} & \text{if } t \text{ is even}. \end{cases}$$

This is the self-conjugate analogue of the weighted core-average conjecture, incorporating the parity of $t$. The supplied text does not state a resolution, so the conjecture is recorded as open.