Coincidence conjecture for squarefree strongly stable ideals

Let $S=\mathbb{K}[x_1,\dots,x_n]$, and let $I\subset S$ be a squarefree strongly stable ideal generated by monomials of degree $d$. Let $I^\tau$ be the monomial ideal obtained from $I$ by the stated squarefree-to-ordinary polarization correspondence. Coincidence conjecture. The following seven numbers coincide:

$$\operatorname{sdepth}_n(I),\quad \operatorname{hdepth}_n(I),\quad \operatorname{hdepth}_1(I),\quad \operatorname{sdepth}_n(I^\tau),\quad \operatorname{hdepth}_n(I^\tau),\quad \operatorname{hdepth}_1(I^\tau),\quad \left\lfloor\frac{H(I^\tau,d+1)}{H(I^\tau,d)}\right\rfloor.$$

This conjecture would unify Stanley depth and Hilbert depth under the squarefree correspondence and reduce these invariants to a Hilbert-function calculation. The supplied text gives no resolution evidence.