Restriction of irreducible modules for $Spi{n}_{2n+1}(K)$ to $Spi{n}_{2n}(K)$

Let $K$ be an algebraically closed field of characteristic $p⩾0$ and let $Y=Spi{n}_{2n+1}(K)$ $(n⩾3)$ be a simply connected simple algebraic group of type $B_n$ over $K\mathrm{.}$ Also let $X$ be the subgroup of type $D_n,$ embedded in $Y$ in the usual way, as the derived subgroup of the stabilizer of a non-singular one-dimensional subspace of the natural module for $Y\mathrm{.}$ In this paper, we give a complete set of isomorphism classes of finite-dimensional, irreducible, rational $KY$-modules on which $X$ acts with exactly two composition factors.