Power containment for the twisted cubic's covariant ideal

Assume $d=3$ and let $r$ be arbitrary, with $3$ not dividing $r$. Let $X\subseteq\mathbf P^3$ be the twisted cubic curve, let $I_X$ be its defining ideal, and let $J_{r,3}$ be the ideal generated by the coefficients of the relevant covariant $\mathfrak G_{r,3}$. Twisted-cubic power-containment conjecture. There is an inclusion

$$(I_X)^r\subseteq J_{r,3},$$

and $r$ is the smallest positive integer $k$ for which $(I_X)^k\subseteq J_{r,3}$. The text notes that $\sqrt{J_{r,3}}=I_X$ and that computations suggest the sharper containment and minimality statement; its status remains open.