Roé-Urbinati's b-divisorial valuation conjecture

Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic $0$, and let $v\in \mathrm{Val}_X$ be a valuation on $X$. For a valuation $v$, let $\mathfrak{a}_\bullet(v)$ denote its associated filtration, and let $Z_X(\mathfrak{a}_\bullet(v))$ be the corresponding b-divisor.

Roé-Urbinati's conjecture. The valuation $v$ is b-divisorial if and only if

$$Z_X(\mathfrak{a}_\bullet(v))\ne 0.$$

This conjecture concerns the characterization of b-divisorial valuations through the b-divisor associated with their valuation filtration. It is known when $\dim X=2$, while the general case is the main motivation of the note and remains open.