The pure-centralizer conjecture for minimal-support elements in Artin groups

Let $B$ be an Artin group with standard generating set ${\mathbf S}$, let ${\mathbf b}\in B$ have minimal length in its conjugacy class, and let $\supp({\mathbf b})$ denote its support. Let $P$ be the pure subgroup and let $\bpi_I:P\to B_I$ be the retraction to the standard parabolic subgroup $B_I$. Pure-centralizer conjecture. If ${\mathbf p}\in P$ centralises ${\mathbf b}$ and

$$\bpi_{\supp({\mathbf b})}({\mathbf p})=1,$$

then ${\mathbf p}$ centralises $B_{\supp({\mathbf b})}$. This conjecture is presented as a generalisation of an earlier proposition and is related to uniqueness of minimal parabolic subgroups; the source does not state that it has been resolved.