Bounded-complement conjecture for Fano type varieties of bounded coregularity

Let $c$ be a nonnegative integer, and let $s(c)$ be the $c$-th Sylvester's number. Let $X$ be a Fano type variety of coregularity at most $c$. Bounded-complement conjecture. If $c=0$, then $X$ admits a $2$-complement; if $c\geq 1$, then $X$ admits a $(2s(c)-3)(s(c)-1)$-complement.

The conjecture gives an explicit complement bound depending only on coregularity, not on the dimension of $X$. It is motivated by coregularity-dependent log canonical threshold bounds and examples of log Calabi–Yau klt varieties with the indicated index; the conjecture remains open.