CW-complex reconstruction from associative higher groupoids

Let $X$ be a CW-complex with sets $I^X_k$ of cells, and let $X$ be an $\infty$-groupoid with $k$-cells $X_k=I^X_k$. For each cell $h e I^X_k$, let $M(h)$ denote its associated manifold diagram and let $\llbracket h\rrbracket$ denote its corresponding framed stratification.

CW reconstruction conjecture. There is a non-unique $\infty$-groupoid $X$ such that $X_k=I^X_k$ and, for each $h e I^X_k$, $M(h)$ is equivalent to $\llbracket h\rrbracket$. Conversely, if $X$ arises from $X$ in this way, then $X$ as a CW-complex can be uniquely recovered from $X$ by a mapping from groupoids to CW-complexes based on the inverse of the generalised Thom–Pontryagin construction.

This is the proposed upper correspondence in the paper's square relating higher groupoids, CW-complexes, manifold diagrams and framed stratifications. It is stated as a conjecture, with no resolution given.