Let $X$ be a normal projective variety over a field of characteristic $p>0$ and $(X, \Delta\geq 0)$ be a pair such that $K_X+\Delta$ is $\mathbb{Q}$-Cartier whose index is not divisible by $p$. Also assume that $A$ is an effective Cartier divisor, then $\tau(X, \Delta)=\tau(X, \Delta+\varepsilon A)$ for $0<\varepsilon\ll 1$. Now my question is, if $(X, \Delta)$ is Strongly $F$-regular does that imply $S^0(X, \tau(X, \Delta)\otimes {\mathcal{O}}_X(M))=S^0(X, \tau(X, \Delta ')\otimes {\mathcal{O}}_X(M))$, where $\Delta ' = \Delta+\varepsilon A$ and $M$ is a Cartier divisor ?

I know one of the inclusion is true for the obvious reason $\Delta '\geq \Delta$, how to prove the other inclusion, if it is true at all!

Thanks