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

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. $$0$$.

• Your equation only contains the product $JK$; the ideals $J$ and $K$ never appear separately. Is this intentional? – David E Speyer Feb 5 at 16:48
• @DavidESpeyer: I am not sure what you meant. One can wrote, let $I=JK$, but being product is crucial for this property to have a hope of being true. – Hailong Dao Feb 5 at 17:13
• Thanks. I've looked up the paper now, and I see that you did mean what you wrote. – David E Speyer Feb 5 at 17:28