The converse conjecture for F-pure thresholds and Segre-product tangent cones

Let $(R,\mathfrak{m})$ be a $d$-dimensional F-finite F-pure non-$\mathbb{Q}$-Gorenstein normal local ring of characteristic $p>0$. Let $\mathrm{c}(\mathfrak{m})$ denote its F-pure threshold, and let $\mathrm{gr}_{\mathfrak{m}}(R)$ be the associated graded ring of $R$ with respect to $\mathfrak{m}$. Converse conjecture. If

$$\mathrm{c}(\mathfrak{m})=d-1,$$

then

$$\mathrm{gr}_{\mathfrak{m}}(R)\cong k[X_{ij}\mid 1\le i\le d-1,\ j=1,2]/(X_{k1}X_{l2}-X_{k2}X_{l1}\mid 1\le k\le l\le d-1).$$

This is the expected converse of the preceding example, which shows that the stated value of the F-pure threshold occurs for the completion of a Segre product that is not $\mathbb{Q}$-Gorenstein. The source provides no resolution of this expectation.