Stanley's squarefree quotient conjecture at Stanley depth d+1

Let $S=K[x_1,\ldots,x_n]$. Let $I\subset S$ be minimally generated by squarefree monomials $f_1,\ldots,f_r$ of degree $d$, and let $E$ be a set of squarefree monomials of degree at least $d+1$; take $J$ to be the monomial ideal generated by $E$. If

$$\sdepth_S(I/J)=d+1,$$

then the squarefree quotient conjecture asserts that

$$\depth_S(I/J)\leq d+1.$$

This is presented as a particular case of Stanley's conjecture for squarefree monomial ideal quotients. The paper studies such cases using polarization and known lower-bound results, but the asserted implication is not given as resolved here.