On the uniform positivity of $F$-signature under reduction modulo $p$

Carvajal-Rojas, Schwede and Tucker asked whether the mod $p$ reductions of a complex klt type singularity have uniformly positive $F$-signature for almost all primes $p$. In this paper, we give an affirmative answer to this conjecture in the case of pure subrings of regular local rings--for example, reductive quotient singularities. We also show that the conjecture can be reduced to the Gorenstein case. Finally, we discuss the connection with $F$-alpha invariants--a characteristic $p$ analog of Tian's alpha invariants introduced by Pande--for log Fano pairs.