Local Converse Theorems for $k$-Middle Supercuspidal Representations of $p$-adic $\text{GL}(kn)$

Let $\text{F}$ be a non-archimedean local field of characteristic zero with residual characteristic $p$. In this paper, we study local converse theorems for $k$-middle supercuspidal representations of $\text{GL}(kn, \text{F})$ via explicit computations of twisted gamma factors on the Galois side, un