The perturbation invariance conjecture for plus test modules

Let $X$ be a normal integral scheme projective over a complete local ring $(R,\mathfrak{m})$ with residual characteristic $p>0$, and let $B$ be a $\mathbb{Q}$-divisor on $X$. Perturbation invariance conjecture. For a positive Cartier divisor $D$ on $X$ and $1\gg\epsilon>0$, one has

$$\tau_+(\omega_X,B)=\tau_+(\omega_X,B+\epsilon D).$$

The conjecture concerns the stability of plus test modules under sufficiently small effective perturbations. The source notes that even the affine case is unclear, although an analogous statement for a variant of perturbed test ideals is known.