Conjecture on deformation to PL affine compactifications

Let $B$ be a compact space with a ${\bf Z}$-affine structure with singularities, with actual and potential singular sets equal, $B^{sing}=B^{pre-sing}$, and assume this set has codimension at least two. Conjecture on deformation to PL affine compactifications. There is a continuous path $\gamma(t)$ of ${\bf Z}$-affine structures with singularities connecting the given structure to the one from a PL compactification, such that $codim(B_t^{pre-sing})\ge2$ for every $t\in[0,1]$ and every $\gamma(t)$ has the Finiteness and Independence properties. This is a proposed deformation principle for affine structures arising in Calabi–Yau collapse; it remains open.