Katthän's weakened Stanley-depth conjecture

Let $S$ be a polynomial ring over a field and let $I\subset S$ be a monomial ideal. Write $\operatorname{depth}$ for homological depth and $\operatorname{sdepth}$ for Stanley depth. Katthän's conjecture. The following two inequalities hold:

$$\operatorname{sdepth}(S/I)\geq\operatorname{depth}(S/I)-1,$$

and

$$\operatorname{sdepth}(I)\geq\operatorname{depth}(I).$$

This is presented as a weaker form of Stanley's conjecture and is stated in the source to be open.