Let $k$ be a field and $S=k[x,y,z]$. Let $m=(x,y,z)$ and $J,K\subseteq m$ be proper homogeneous ideals in $S$. Is this true that we always have:

$$[JK:(x)][JK:(y,z)]\subseteq JK \ ?$$

**Some background**: the case $K=m$ was asked at Is $[Im:(x)][Im:(y,z)]\subseteq Im$ in $k[x,y,z]$?. It was not so hard. One can prove that the result holds if $J,K$ are monomial or if $K\subset m^2$ is integrally closed. The statement is related to the question of whether the product of two ideals is Golod. See the preprint Dao and De Stefani - On monomial Golod ideals for some partial results and why a modest monetary reward is offered for this frustrating problem (Question 4.6). My personal guess is that this is probably true in char. $0$.