Castelnuovo-Mumford regularity bound for log canonical projective schemes

Let $R=k[x_0,\ldots,x_n]$, and let $I=(f_1,\ldots,f_t)$ be a homogeneous ideal generated in degrees $d_1\geq d_2\geq \cdots\geq d_t\geq 1$, of codimension $r$. Set $X=\operatorname{Proj} R/I$, and assume that, except for some isolated points, $X$ is local complete intersection log canonical and $\dim X\geq 1$. Regularity-bound conjecture. Then

$$\operatorname{reg} R/I\leq (\dim X+1)!\left(\sum_{i=1}^{r}d_i-r\right).$$

The conjecture proposes a sharper Castelnuovo-Mumford regularity bound under weaker assumptions allowing isolated non-log-canonical points; the paper explains that its method does not establish the required singularity statement for the residual intersection.