Möbius function and homotopy type conjecture for weak order on monotone triangles

Let $T'$ and $T$ be monotone triangles with $T'\leq_W T$, and let $\Delta(T',T)$ denote the order complex of their open interval in $<_W$. Write $\operatorname{Des}(T)$ for the descent set of $T$, let $w_0(J)$ be the longest element associated with $J\subset\operatorname{Des}(T)$, and define

$$J:=\{m:T'_m\neq T_m\}.$$

Homotopy-type conjecture. The complex $\Delta(T',T)$ is contractible unless

$$T'=T\times\pi_{w_0(J)},$$

in which case $\Delta(T',T)$ is homotopy equivalent to a $(\#J-2)$-dimensional sphere.

This conjecture would give a simple description of the homotopy types, and hence the Möbius function, of open intervals in the weak order on monotone triangles, paralleling the corresponding description for weak order on permutations. The supplied text does not indicate whether the claim has been proved or remains open.