Conjecture on regularity bounds for squarefree powers of simplicial trees

Let $\Delta$ be a simplicial tree of dimension $t-1$, let $I(\Delta)$ denote its facet ideal in the polynomial ring $R$, and let $\nu_1(\Delta)$ and $\nu(\Delta)$ denote its induced matching number and matching number, respectively. The squarefree power $I(\Delta)^{[k]}$ is defined for $1\leq k\leq \nu(\Delta)$. Regularity conjecture.

$$k-1+(t-1)\nu_1(\Delta) \leq \operatorname{reg}\left(\frac{R}{I(\Delta)^{[k]}} \right)\leq k-1+(t-1)\nu(\Delta)$$

for all $1\leq k\leq \nu(\Delta)$. This would extend the expected regularity bounds for squarefree powers of edge ideals to simplicial trees; the preceding proposition establishes the lower bound, while the upper bound is the conjectural part.