The stable square-length asymptotic formula for orthogonal characters

Let $1\ne w\in\mathbb{\mathbf{F}}$, let $\mu=(\mu_{1},\ldots,\mu_{k})$ be a partition, and let $e_{w,\mu}$ be the number of stable-square-length-extremal surfaces, orientable or non-orientable, with non-oriented boundary components corresponding to $w^{\mu_{1}},\ldots,w^{\mu_{k}}$, modulo the equivalence specified in the source. Let $\nu$ be a partition, let $\psi^{\nu[N]}$ be the corresponding stable irreducible character of $\mathrm{O}(N)$, and let $b_{\nu,\mu}$ be the coefficient of $\zeta_{\mu}$ in its power-sum expansion. Orthogonal stable square-length conjecture. For every partition $\nu$,

$$\mathbb{E}_{w}\left[\psi^{\nu[N]}\right]=N^{-\lvert\nu\rvert\mathrm{ssql}(w)}\left(\left(\sum_{\mu\vdash\lvert\nu\rvert}b_{\nu,\mu}e_{w,\mu}\right)+O(N^{-1})\right).$$

The conjecture extends the stable square-length asymptotic picture to orthogonal groups, where the relevant extremal surfaces may be non-orientable; the source notes exact surface formulas for power sums but does not prove this character formula.