The multiplier–test ideal comparison conjecture

Let $Y$ be a smooth, irreducible variety over an algebraically closed field $k$ of characteristic zero, and let $\mathfrak{a}$ be a nonzero ideal on $Y$. Given any model $Y_A$ and $\mathfrak{a}_A$ for $Y$ and $\mathfrak{a}$ over a subring $A$ of $k$, finitely generated over $\mathbf{Z}$, let $Y_s$ and $\mathfrak{a}_s$ denote the corresponding fibers at a closed point $s\in\operatorname{Spec} A$. The ideals $\mathcal{J}(Y,\mathfrak{a}^{\lambda})$ and $\tau(Y_s,\mathfrak{a}_s^{\lambda})$ are respectively the multiplier and generalized test ideals. The multiplier–test ideal comparison conjecture. There is a dense set of closed points $S\subset\operatorname{Spec} A$ such that

$$\mathcal{J}(Y,\mathfrak{a}^{\lambda})_s=\tau(Y_s,\mathfrak{a}_s^{\lambda})$$

for every $\lambda\in\mathbf{R}_{\geq 0}$ and every $s\in S$. The conjecture seeks equality between characteristic-zero multiplier ideals and reductions of test ideals simultaneously for all exponents; the source notes that equality for each fixed exponent is already known on an open subset, while the uniform dense-set assertion is the conjectural part.