The reflection deletion conjecture for Coxeter groups

Let $W$ be a Coxeter group, let $w\in W$, and let $t\in W$ be a reflection. Write $\ell$ for the Coxeter length function.

Reflection deletion conjecture. If

$$\ell(wt)<\ell(w)-1,$$

then there exists a reduced expression

$$w=s_{i_1}\dotsm s_{i_{\ell-1}}s_{i_\ell}s_{i_{\ell+1}}\dotsm s_{i_r}$$

with $i_{\ell-1}=i_{\ell+1}=i$ such that

$$wt=s_{i_1}\dotsm s_i\hat{s}_{i_\ell}s_i\dotsm s_{i_r},$$

where the factor $s_{i_\ell}$ has been removed. The conjecture is a Coxeter-group reformulation of the preceding root-order conjecture and remains open in the supplied source.