A non-Levi branching rule in terms of Littelmann paths

We prove a conjecture of Naito-Sagaki about a branching rule for the restriction of irreducible representations of $\mathfrak{s}\mathfrak{l}(2n,\mathbb{C})$ to $\mathfrak{s}\mathfrak{p}(2n,\mathbb{C})$. The conjecture is in terms of certain Littelmann paths, with the embedding given by the folding of the type $A_{2n-1}$ Dynkin diagram.