Torus-bundle conjecture over controlled Riemannian orbifolds

Let $D\geq1$, $m,l\in\mathbb N$, and $\iota>0$. Let $(M,g)\in\mathcal{M}_{\rm Rc}(m,D)$ and let $(X,d_X)$ be an $(l,\iota)$-controlled $k$-dimensional Riemannian orbifold. Assume

$$d_{\rm GH}(M,X)<\varepsilon$$

and

$$b_1(M)-b_1(X)=m-k.$$

Torus-bundle conjecture. There exists $\varepsilon=\varepsilon(m,l,\iota)>0$ such that these assumptions imply that $M$ is a torus bundle over $X$. This is presented as an open problem combining collapsing geometry with Ricci-flow smoothing; it seeks a global bundle conclusion when the Betti-number drop equals the collapsing dimension.