Simply laced Bruhat preclosure closure conjecture

Let $W$ be a Coxeter group of type $A$, $D$, or $E$, let $T$ be its set of reflections, and let $A\subseteq T$. Write $\overline{A}$ for the Bruhat preclosure of $A$.

Simply laced Bruhat preclosure conjecture. The Bruhat preclosure is idempotent:

$$\overline{A}=\overline{\overline{A}}.$$

The conjecture is motivated by computational evidence in types $B_n$ and $D_n$ and by the failure of the closure property in types $H$ and $F$. The paper proves that the preclosure is a closure in type $A$, but the stated claim for all types $A$, $D$, and $E$ remains open.