On the strongly regular locus of the inertia stack of $\mathrm{Bun}_G$

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 $\operatorname{Bun}_G^b$ is open in the inertia stack of $\operatorname{Bun}_G$.

As a consequence, we extend the computation of Hansen--Kaletha--Weinstein of trace distributions of the cohomology of local shtuka spaces $\mathrm{Sht}_{G,b,\mu}$ 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.