The local converse theorem for quasi-split $O_{2n}$ and $S{O}_{2n}$

Let $F$ be a non-archimedean local field of characteristic not equal to 2. In this paper, we prove the local converse theorem for quasi-split ${\text{Ø}}_{2n}(F)$ and ${\text{SO}}_{2n}(F)$, via the description of the local theta correspondence between ${\text{Ø}}_{2n}(F)$ and ${\text{Sp}}_{2n}(F)$. More precisely, as a main step, we explicitly describe the precise behavior of the $\gamma$-factors under the correspondence. Furthermore, we apply our results to prove the weak rigidity theorems for irreducible generic cuspidal automorphic representations of ${\text{Ø}}_{2n}(\text{A})$ and ${\text{SO}}_{2n}(\mathbb{A})$, respectively, where $\text{A}$ is a ring of adele of a global number field $L$.