Jöllenbeck's strong gcd-condition conjecture for Golod monomial rings

Let $S=k[x_1,\ldots,x_n]$, let $\mathfrak a=\langle m_1,\ldots,m_l\rangle\subset S$ be a monomial ideal, and set $A=S/\mathfrak a$. The ideal $\mathfrak a$ satisfies the strong gcd-condition if it has the combinatorial property defined in the source. Jöllenbeck's strong gcd-condition conjecture. The quotient $A$ is Golod if and only if $\mathfrak a$ satisfies the strong gcd-condition. In particular, Golodness is independent of the characteristic of $k$. This conjecture would characterize Golod monomial rings by a purely combinatorial condition. The supplied text gives several classes for which the implication to Golodness follows under property (P), but does not report a proof of the full equivalence.