The first author's conjecture on the stable abelianization of $\mathfrak{h}_{g,1}$

Let H_\mathfrak{h}_{g,1} be the Lie algebra appearing in the lower homomorphism described in the paper, and let $H_1(\mathfrak{h}_{g,1})$ denote its first homology group. The stable abelianization is obtained by taking the direct limit as $g$ increases.

Stable abelianization conjecture. The stable abelianization of the Lie algebra $\mathfrak{h}_{g,1}$ vanishes; namely

$$\lim_{g\to\infty} H_1(\mathfrak{h}_{g,1})=0.$$

This predicts that no first homology survives in the stable range. The preceding discussion reports that the related conjecture asserting that the lower homomorphism induces an isomorphism on $H_1$ was disproved by Conant, Kassabov and Vogtmann; the stable-vanishing conjecture is presented as a remaining expectation supported by known results and computations.