On the strongly regular locus of the inertia stack of ${\mathrm{B}\mathrm{u}\mathrm{n}}_G$

Let $G$ be a connected reductive group over a finite extension of ${\mathbb{Q}}_p$. We show that for each $b\in B(G)$, the strongly regular locus of the inertia stack of ${\mathrm{Bun}}_G^b$ is open in the inertia stack of ${\mathrm{Bun}}_G$. As a consequence, we extend the computation of Hansen–Kaletha–Weinstein of trace distributions of the cohomology of local shtuka spaces ${\mathrm{S}\mathrm{h}\mathrm{t}}_{G,b,\mu }$ to non-basic $b$. If $b$ is closed in $B(G,\mu )$, or $b$ is basic and has only one specialization in $B(G,\mu )$, then we compute the trace distribution of the entire strongly regular locus. In the process, we prove some results on the behavior of characteristic classes under cohomologically smooth pullback.