Brenti's conjecture on special matchings and Kazhdan–Lusztig $R$-polynomials

Let $W$ be a Coxeter group of type $A$, let $u,v\in W$, and let $M$ be a special matching of the Bruhat interval $[u,v]$. For $x,y\in [u,v]$ with $x\le y$, write $R_{x,y}(q)$ for the Kazhdan–Lusztig $R$-polynomial. Brenti's conjecture. For all such $x,y$, one has

$$R_{x,y}(q)=\begin{cases} R_{M(x),M(y)}(q) & \operatorname{if } M(x)\lhd x,\ M(y)\lhd y,\\ R_{M(x),M(y)}(q) & \operatorname{if } M(x)\rhd x,\ M(y)\rhd y,\\ (q-1)R_{x,M(y)}(q)+qR_{M(x),M(y)}(q) & \operatorname{if } M(x)\rhd x,\ M(y)\lhd y,\\ q^{-1}R_{M(x),M(y)}(q)+(q^{-1}-1)R_{M(x),y}(q) & \operatorname{if } M(x)\lhd x,\ M(y)\rhd y. \end{cases}$$

The conjecture extends Brenti's result for intervals starting from the identity to arbitrary Bruhat intervals in type $A$. The paper's abstract says that this conjecture is proved, so its status is recorded as solved; the displayed assertion is retained as the source statement.