Amdeberhan's conjectured q-series identity involving Riordan numbers

Let $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_r)$ be a partition, and define

$$g_\lambda(q)=\prod_{j=1}^r \frac{q^{\lambda_j}}{1+q^{\lambda_j}}.$$

For a partition $\lambda$, let $z_\lambda$ be the standard symmetric-function multiplicity factor; retain the notation $\operatorname{Ev}(\lambda)$, $\ell(\tilde{\lambda})$, and $\chi^\mu_{\tilde{\lambda}}$, and let $R_{2N+1}(2n)$ and $R^c_{2N}(2n)$ denote the partition sets specified in the paper. Amdeberhan's conjecture. For all integers $N\geq 1$,

$$\sum_{n\geq 0}\sum_{\lambda\vdash n}\frac{g_\lambda(q)}{z_\lambda}\sum_{\tilde{\lambda}\in\operatorname{Ev}(\lambda)}\sum_{\mu\in R_{2N+1}(2n)}(-1)^{\ell(\tilde{\lambda})}\chi^\mu_{\tilde{\lambda}}=\sum_{n\geq 0}\sum_{\lambda\vdash n}\frac{g_\lambda(q)}{z_\lambda}\sum_{\tilde{\lambda}\in\operatorname{Ev}(\lambda)}\sum_{\mu\in R^c_{2N}(2n)}\chi^\mu_{\tilde{\lambda}}.$$

The identity is attributed to Tewodros Amdeberhan and is motivated by an earlier question; the paper reports finite-order verification for examples, including $N=1$, but does not establish the conjecture in general.