The contraction conjecture for the weight-two abelianization of symplectic derivations

Let $H_n$ be the underlying symplectic vector space, let $T(H_n)$ be its tensor algebra, and let $\mathrm{Der}^+(T(H_n))$ denote the positively graded derivations of $T(H_n)$. Write $H_1(\mathrm{Der}^+(T(H_n)))_2$ for the weight-two summand of its abelianization, and consider the contraction

$$C_{13}\colon \mathrm{Der}(T(H_n))(2)\longrightarrow H_n^{\otimes 2}.$$

Contraction conjecture. The contraction $C_{13}$ induces an isomorphism

$$H_1(\mathrm{Der}^+(T(H_n)))_2\cong H_n^{\otimes 2}.$$

The preceding proposition establishes only that $C_{13}$ induces a surjection from the weight-two abelianization onto $H_n^{\otimes 2}$; the asserted injectivity, and hence the isomorphism, is the conjectural part. This describes the full weight-two contribution to the abelianization of the Lie algebra of positive derivations.