Morita's trace-pullback conjecture for the homology-cobordism group

Let $\mathcal H_{g,1}$ and $\overline{\mathcal H}_{g,1}$ be the homology-cobordism groups in the source, let $\sigma$ be the homomorphism to the inverse limit of automorphism groups of nilpotent quotients, and let $\bar\sigma$ be its induced homomorphism on $\overline{\mathcal H}_{g,1}$. For each $k\geq1$, let $\tilde t_{2k+1}$ be the corresponding trace class in the second cohomology of that inverse limit. Morita's conjecture. For every $k$, $\bar\sigma^*(\tilde t_{2k+1})$ is non-trivial in $H^2(\overline{\mathcal H}_{g,1})$, while $\sigma^*(\tilde t_{2k+1})$ is trivial in $H^2(\mathcal H_{g,1})$. The assertion is proposed as a way to obtain additive invariants of the group of homology $3$-sphere cobordism classes and remains open.