Fiber homeomorphism conjecture for maps to totally nonnegative spaces

Let $f_{(i_1,\dots,i_d)}$ be one of the maps considered in the paper, let $p\in Y_w^o$, and let $\Delta((i_1,\dots,i_d),w)$ denote the corresponding subword complex. Consider the cell decomposition of $f^{-1}_{(i_1,\dots,i_d)}(p)$ induced by that of a simplex. Fiber homeomorphism conjecture. There is a cell-structure-preserving homeomorphism from this cell decomposition to the interior dual block complex of $\Delta((i_1,\dots,i_d),w)$. In particular, $f^{-1}_{(i_1,\dots,i_d)}(p)$ is contractible. The claim is presented as a consequence of the preceding regular-CW conjecture together with the proven face-poset and contractibility results; its resolution is not supplied in the excerpt.