Optimal second-leading-term conjecture for stable orbital integrals on general linear Lie algebras

Let $F$ be a non-Archimedean local field, let $\boldsymbol{o}$ be its ring of integers with residue field $\boldsymbol{\text{}}$ of cardinality $q$, and let $\gamma\in\mathfrak{gl}_n(\boldsymbol{o})$ be regular semisimple with $n\geq3$. Use the notation $B(\gamma)$, $d_i$, $\bar d_{\gamma_i}$, $\rho(\gamma)$, $S(\gamma_i)$, and $r_i$ from the paper, and let $\alpha(\bar d_{\gamma_i})$ equal $0$ for $\bar d_{\gamma_i}=0$ or $1$, $1$ for $\bar d_{\gamma_i}=2$, and $2$ for $\bar d_{\gamma_i}\geq3$. The optimal second-leading-term conjecture. If $\operatorname{char}(F)=0$ or $\operatorname{char}(F)>n$, then

$$\mathcal{SO}_{\gamma}=\frac{\#\operatorname{GL}_{n}(\kappa)}{q^{n^{2}}}\prod_{i\in B(\gamma)}\frac{q^{d_i}}{q^{d_i}-1}\left(1+\alpha(\bar d_{\gamma_i})q^{-d_i}+O(q^{-2d_i})\right),$$

and

$$\mathcal{SO}_{\gamma,d\mu}=q^{\rho(\gamma)}\prod_{i\in B(\gamma)}\left(q^{S(\gamma_i)d_i}+q^{(S(\gamma_i)-1)d_i}+\cdots+q^{(S(\gamma_i)-r_i+1)d_i}\right)\left(1+\alpha(\bar d_{\gamma_i})q^{-d_i}+O(q^{-2d_i})\right).$$

Equivalently, the term $\alpha(\bar d_{\gamma_i})x^{-1}$ in the description of $N'_{\gamma_i,d\mu}(x)$ is optimal. The conjecture is established for $n=3$ and when $S(\gamma_i)=d_{\gamma_i}=2$, but remains open in general.