4 problems

Three-factor radical-generation conjecture. The radical of $J_{p,3}$ is the prime ideal $I_{p,3}$:

Two-factor minimal-generation conjecture. The ideal $I_{p,2}$ is minimally generated by

Linear-eliminant membership conjecture. If $m\geq 2$, $p\geq 2m+2$, and $R\cup C\neq\bar R\cup\bar C$, then

Tightness conjecture. The bound from Theorem bound-alternate is tight for all such $p$, $k$, and $r$; equivalently,