The subgroup-fixed-locus compatibility conjecture for Iritani isomorphisms

Let $X$ be a variety with a $G$-action, and let $H_1\triangleleft H_2<G$ and $H_1'\triangleleft H_2'<G$ satisfy

$$H_1<H_1'<H_2'<H_2<G.$$

The corresponding fixed loci of maximal F-bundle bases are

$$(B_{[X/H_1]})^{H_2}\hookrightarrow (B_{[X/H_1']})^{H_2'}.$$

Subgroup-fixed-locus compatibility conjecture. These inclusions commute with Iritani's partial isomorphisms and with the action of $\mathsf{MT}$. This generalizes the preceding compatibility conjectures to the diagram of subgroup-fixed maximal F-bundles; the source does not state that it is proved.