Distinguished Representations for ${\mathrm{S}\mathrm{L}}_n(D)$ where $D$ is a quaternion division algebra over a $p$-adic field

Let $D$ be a quaternion division algebra over a non-archimedean local field $F$ of characteristic zero. Let $E\mathrm{/}F$ be a quadratic extension and ${\mathrm{S}\mathrm{L}}_n^{\ast }(E)={\mathrm{G}\mathrm{L}}_n(E)\cap {\mathrm{S}\mathrm{L}}_n(D)$. We study distinguished representations of ${\mathrm{S}\mathrm{L}}_n(D)$ by the subgroup ${\mathrm{S}\mathrm{L}}_n^{\ast }(E)$. Let $\pi$ be an irreducible admissible representation of ${\mathrm{S}\mathrm{L}}_n(D)$ which is distinguished by ${\mathrm{S}\mathrm{L}}_n^{\ast }(E)$. We give a multiplicity formula, i.e. a formula for the dimension of the $\mathbb{C}$-vector space ${\mathrm{H}\mathrm{o}\mathrm{m}}_{{\mathrm{S}\mathrm{L}}_n^{\ast }(E)}(\pi ,\text{mathbbm}1)$, where $\text{mathbbm}1$ denotes the trivial representation of ${\mathrm{S}\mathrm{L}}_n^{\ast }(E)$. This work is a non-split inner form analog of a work by Anandavardhanan-Prasad which gives a multiplicity formula for ${\mathrm{S}\mathrm{L}}_n(F)$-distinguished irreducible admissible representation of ${\mathrm{S}\mathrm{L}}_n(E)$.