The stable square-length asymptotic formula for unitary characters

Let $1\ne w\in\mathbb{\mathbf{F}}$, let $\mu=(\mu_{1},\ldots,\mu_{k})$ be an integer sequence with no zero entries, and let $d_{w,\mu}$ be the number of orientable stable-square-length-extremal surfaces with oriented boundary components corresponding to $w^{\mu_{1}},\ldots,w^{\mu_{k}}$, modulo the equivalence specified in the source. Let $\lambda=(\lambda^{+},\lambda^{-})$ be a pair of partitions, let $\xi^{\lambda[N]}$ be the corresponding stable irreducible character of $\mathrm{U}(N)$, and let $a_{\lambda,\mu}$ be the coefficient of $\zeta_{\mu}$ in its power-sum expansion. Unitary stable square-length conjecture. For every such $w$, $\mu$ and $\lambda$,

$$\mathbb{E}_{w}\left[\xi^{\lambda[N]}\right]=N^{-\lvert\lambda\rvert\mathrm{ssql}(w)}\left(\left(\sum_{\mu:\lvert\mu\rvert=\lvert\lambda\rvert}a_{\lambda,\mu}d_{w,\mu}\right)+O(N^{-1})\right).$$

The conjecture refines the proposed role of stable square length for arbitrary stable unitary representations by identifying the leading coefficient with extremal orientable surfaces. The source presents this as conjectural despite related exact surface expansions for power sums.