F-threshold of determinantal rings

In this paper, by using a combinatorial approach, we establish a new upper bound for the F-threshold $c^{\text{mm}}(\text{mm})$ of determinantal rings generated by maximal minors. We prove that $c^{\text{mm}}(\text{mm})$ coincides with the $a$-invariant in the case of $3\times n$ and $4\times n$ matrices and we conjecture such equality holds for all matrices.