Conjecture on the representation structure of electrical Lie algebras of type D

Let $\mathfrak{sp}_{2n}$ be the symplectic Lie algebra, let $V_{\nu}$ be its standard representation, let $V_{\mathbf{0}}$ be its trivial representation, and let $V_{\lambda}$ be the irreducible $\mathfrak{sp}_{2n}$-representation whose highest weight $\lambda$ is the sum of the first and second fundamental weights. Representation-structure conjecture. The electrical Lie algebra $\mathfrak{e}_{D_{2n+1}}$ is an extension of

$$\mathfrak{sp}_{2n}\ltimes V_{\nu}$$

by $V_{\lambda}\oplus V_{\mathbf{0}}$; equivalently, there is a short exact sequence

$$0\longrightarrow V_{\lambda}\oplus V_{\mathbf{0}}\longrightarrow \mathfrak{e}_{D_{2n+1}}\longrightarrow \mathfrak{sp}_{2n}\ltimes V_{\nu}\longrightarrow 0.$$

This conjecture refines the preceding structural results, including the radical description and quotient by the radical, by specifying the remaining extension as an $\mathfrak{sp}_{2n}$-representation. The source gives no evidence of a resolution.