Uniform reduction conjecture for test ideals and multiplier ideals

Let $R$ be a domain smooth over ${\mathbf Z}$ with $R\otimes_{\mathbf Z}{\mathbf Q}\neq 0$, let $f\in R$ be nonzero, and for each prime $p$ let $R_p=R\otimes_{\mathbf Z}{\mathbf F}_p$ and let $f_p$ denote the image of $f$. Let $\mathcal{J}(f^{\lambda})$ be the multiplier ideal of $f^{\lambda}$ and $\tau(f_p^{\lambda})$ its test ideal after reduction modulo $p$. Uniform reduction conjecture. With the notation of Theorem 1, there are infinitely many primes $p$ such that for all $\lambda\in{\mathbf R}_+$,

$$\tau(f_p^{\lambda})=\mathcal{J}(f^{\lambda})_p.$$

This strengthens the Hara–Yoshida comparison between multiplier ideals and test ideals, which gives equality for every fixed $\lambda$ and all sufficiently large primes depending on $\lambda$. The conjecture asks for infinitely many primes for which the equality holds simultaneously for every positive real exponent; its resolution is not indicated in the supplied text.