Morita's basis conjecture for the symplectic second cohomology of the derivation Lie algebra

Let $\mathfrak h_{g,1}^{\mathbb Q}$ be the graded Lie algebra under consideration, and let $\operatorname{Sp}$ act on it through the symplectic action on $H_{\mathbb Q}$. Define $e_1,t_3,t_5,\ldots\in H^2(\mathfrak h_{g,1}^{\mathbb Q})^{\operatorname{Sp}}$ by the higher intersection pairings and the trace maps described in the preceding definition. Morita's conjecture. The classes $e_1,t_3,t_5,\ldots$ are all non-trivial, are linearly independent, and form a basis of

$$H^2(\mathfrak h_{g,1}^{\mathbb Q})^{\operatorname{Sp}}.$$

The claim is proposed as the expected description of the symplectic-invariant second cohomology and remains open.