Abelianization conjecture for the stable symplectic derivation Lie algebra

Let ${\mathfrak h}^{\mathbb Q}_{g,1}$ be the rational graded Lie algebra of symplectic derivations, let $H_{\mathbb Q}$ be the rational symplectic module, and let ${\mathfrak h}^{\mathbb Q}_\infty$ be its stable direct limit. Abelianization conjecture. The maps given by the first Johnson homomorphism and the traces induce an isomorphism

$$H_1({\mathfrak h}^{\mathbb Q}_{g,1})={\mathfrak h}^{\mathbb Q}_{g,1}/[{\mathfrak h}^{\mathbb Q}_{g,1},{\mathfrak h}^{\mathbb Q}_{g,1}]\cong {\Lambda}^3 H_{\mathbb Q}\oplus\biggl(\bigoplus_{k\geq 1} S^{2k+1}H_{\mathbb Q}\biggr),$$

and an analogous statement holds for ${\mathfrak h}^{\mathbb Q}_\infty$. The conjecture is motivated by explicit low-degree computations, but the source provides no proof or resolution.