Kummini–Lakshmibai–Sastry–Seshadri regularity conjecture for Grassmannian Schubert patches

Let $B\subseteq GL_n(\mathbb{C})$ be the Borel subgroup of upper triangular matrices, and let $w\in S_n$ be a Grassmannian permutation with descent at position $k$. Let $Y$ be the standard open patch associated with $w$, let $J_w$ be the prime ideal generated by the essential minors defining the corresponding patch $Y_w$, and suppose that for some $r\in[k-1]$ one has

$$w_{k-r+i}=n-k+i\quad\text{for all }i\in[r]$$

and $w_1=1$. Define $\widetilde w$ by $(\widetilde w_1,\ldots,\widetilde w_k)=(n-w_k+1,\ldots,n-w_1+1)$, write

$$(\widetilde w_1,\ldots,\widetilde w_k)=(k-r+1,k-r+2,\ldots,k,a_{r+1},\ldots,a_{n-1},n),$$

set $a_r=k$ and $a_k=n$, and, for $r\leq i\leq k-1$, define $m_i=a_{i+1}-a_i$. Kummini–Lakshmibai–Sastry–Seshadri conjecture. The regularity should satisfy

$$\operatorname{reg}(\mathbb{C}[Y]/J_w)=\sum_{i=r}^{k-1}(m_i-1)i.$$

The paper states that this conjecture is false and provides a counterexample, replacing it with an alternate explicit combinatorial formula for the regularity.