Orbifold dimensional reduction conjecture for character stacks

Let $M$ be a Seifert-fibred $3$-manifold, let $G$ be a semisimple group, and for $n>0$ let $\Sigma_{\mathrm{orb},M}^{[n]}$ be the quotient of $M$ by the $S^1$-action composed with the $n$-th multiplication map, with generic stabilizer $\mathbb{Z}/n\mathbb{Z}$. Let $\iota_n:\mathcal{L}\mathrm{oc}_G(\Sigma_{\mathrm{orb},M}^{[n]})\to\mathcal{L}\mathrm{oc}_G(M)$ and $\bar\iota_n:\mathrm{Loc}_G(\Sigma_{\mathrm{orb},M}^{[n]})\to\mathrm{Loc}_G(M)$ be the induced maps. Orbifold dimensional reduction conjecture. (i) The character stack $\mathcal{L}\mathrm{oc}_G(\Sigma_{\mathrm{orb},M}^{[n]})$ admits a canonical $0$-shifted symplectic structure. (ii) The map $\iota_n$ admits a natural Lagrangian structure. (iii) For a sufficiently divisible $N$ as in the preceding corollary, there is a natural isomorphism

$$\bar\iota_{N,*}\mathcal{BPS}_{\mathrm{Loc}_G(\Sigma_{\mathrm{orb},M}^{[N]})}^{(0)}\cong\mathcal{BPS}_{\mathrm{Loc}_G(M)}^{(0)}.$$

The conjecture is motivated by support containment for BPS sheaves on Seifert-fibred $3$-manifolds; no resolution status is given in the supplied text.