Stronger variable reduction conjecture for monomial ideals with linear quotients

Let $K$ be a field, let $S=K[x_1,x_2,\ldots,x_n]$, let $I\subset S$ be a proper monomial ideal with linear quotients, and write $[n]=\\{1,\ldots,n\\}$. The stronger variable reduction conjecture. There exists $i\in[n]$ such that

$$\operatorname{depth}(S/(I,x_i))\geq\operatorname{depth}(S/I)$$

and

$$\operatorname{sdepth}(S/(I,x_i))\leq\operatorname{sdepth}(S/I).$$

This strengthens the preceding depth-preserving variable conjecture by adding a Stanley-depth inequality; the source gives no resolution.