Equality-case integral-closure conjecture for multiplicity bounds

Let $R$, $\mathfrak{a}$ and $J$ be as in the relevant corollary, with positive integers $a_1,\ldots,a_d,b_1,\ldots,b_d$ supplied by that setup. Suppose that

$$e(\mathfrak{a})=\left(\frac{d}{\operatorname{c}_{-}^J(\mathfrak{a})}\right)^d e(J).$$

Put $b_1/a_1=\cdots=b_d/a_d=t/s$, where $s$ and $t$ are positive integers. Equality-case conjecture. Then $\mathfrak{a}^t$ and $J^s$ have the same integral closure. The source notes that this is known when $d=2$, and more generally when $\{a_1,\ldots,a_d\}$ has at most two elements; the general case remains open in the stated context.