Amdeberhan's character table conjecture

Let $\lambda \vdash n$. Write $Ev(\lambda)$ for the set of even refinements of $\lambda$, let $\mathcal{R}_{2N+1}(2|\lambda|)$ and $\mathcal{R}^{c}_{2N}(2|\lambda|)$ be the partition sets appearing in the conjecture, and let $\chi^{\mu}_{\tilde{\lambda}}$ denote the irreducible character value of the symmetric group indexed by $\mu$ on the conjugacy class indexed by $\tilde{\lambda}$. Amdeberhan's character table conjecture. If $\lambda \vdash n$, then

$$\sum_{\tilde{\lambda} \in Ev(\lambda),\, \mu \in \mathcal{R}_{2N+1}(2|\lambda|)}(-1)^{\ell(\tilde{\lambda})}\chi^{\mu}_{\tilde{\lambda}} = \sum_{\tilde{\lambda} \in Ev(\lambda),\, \mu \in \mathcal{R}^{c}_{2N}(2|\lambda|)}\chi^{\mu}_{\tilde{\lambda}}.$$

This identity is presented as one of Amdeberhan's conjectures, arising from a broader conjecture on $q$-series. The notation for the sets $Ev(\lambda)$, $\mathcal{R}_{2N+1}(2|\lambda|)$, and $\mathcal{R}^{c}_{2N}(2|\lambda|)$ is not defined in the supplied context, so the conjecture's scope and status require verification.