  Let $D$ denote a quaternion division algebra over a non-archimedean local
field $F$ with characteristic zero. Let $Sp_n(D)$ be the unique non-split inner
form of the symplectic group $Sp_{2n}(F)$. An irreducible admissible
representation $(\pi, V)$ of $GL_{n}(D)$ is said to have a symplectic model (or
said to be $Sp_n(D)$-distinguished) if there exists a linear functional $\phi$
on $V$ such that $\phi(\pi(h)v) = \phi(v)$ for all $v \in V$ and $h \in
Sp_n(D)$. This article classifies those ladder representations of $GL_n(D)$
that possess a symplectic model (i.e., those representations that are
$Sp_n(D)$-distinguished). Recently, Prasad conjectured that non-supercuspidal
discrete series representations of $GL_n(D)$ do not admit a symplectic model.
We confirm this for the Steinberg representations, which serve as canonical
examples of discrete series representations. Furthermore, we demonstrate the
hereditary nature of the symplectic model for induced representations derived
from finite-length representations. In addition, we prove a part of Prasad's
conjecture, which provides a family of irreducible unitary representations, all
equipped with a symplectic model.
