The converse conjecture for symbolic F-thresholds of radical ideals

Let $(R,\mathfrak{m})$ be an $F$-finite regular local ring and let $I$ be an unmixed radical ideal. The ideal $I$ is symbolic $F$-split when it has the symbolic $F$-splitting property, and $\mathcal{C}^{\mathfrak{m}}(I^{(\bullet)})$ denotes the symbolic $F$-threshold of $I$ with respect to $\mathfrak{m}$.

Converse conjecture. If $I$ is symbolic $F$-split, then

\mathcal{C}^{\mathfrak{m}}(I^{(\bullet)})=\operatorname{\big\text{-}height}(I).

The paper establishes that attaining the big-height bound implies that a radical ideal is unmixed and symbolic $F$-split; this conjecture asserts the converse implication. The statement concerns when symbolic $F$-splitting characterizes equality in the upper bound for symbolic $F$-thresholds.