The split case of the Prasad--Takloo-Bighash conjecture for cuspidal representations of level zero

Let $E/F$ be a quadratic extension of non archimedean local fields of odd residual characteristic. We prove a conjecture of Prasad and Takloo-Bighash, in the case of cuspidal representations of depth zero of $\mathrm{GL}(2m,F)$. This conjecture characterizes distinction for the pair $(\mathrm{GL}(2m,F),\mathrm{GL}(m,E))$ with respect to a character $\mu\circ \mathrm{det}$ of $\mathrm{GL}(m,E)$, in terms of certain conditions on Langlands paremeters, including an epsilon value. We also compute the multiplicity of the involved equivariant linear forms when $E/F$ is unramified, and also when $\mu$ is tame. In both cases this multiplicity is at most one.