The Morita-pairing conjecture for secondary Borel classes

Let $\mu_k\in H_{4k}(\mathrm{Out}\,F_{2k+2};\mathbb{Q})$ be the Morita class, and let $\overset{\circ}{\beta}_{2k+1}$ be the secondary class defined at the critical rank $2k+2$, with its stated indeterminacy. Via the natural lift of $\mu_k$ to $H_{4k}(\mathrm{Aut}\,F_{2k+2};\mathbb{Q})$, the pairing $\langle\overset{\circ}{\beta}_{2k+1},\mu_k\rangle$ is defined. Morita-pairing conjecture. For a suitable choice of $\overset{\circ}{\beta}_{2k+1}$ within its indeterminacy,

$$\langle\overset{\circ}{\beta}_{2k+1},\mu_k\rangle\neq 0.$$

This conjecture proposes a geometric detection of the Morita classes by secondary characteristic classes; its general status is open, and the value can be independent of the indeterminacy when the projection of $\mu_k$ to $H_{4k}(\mathrm{GL}(2k+2,\mathbb{Z});\mathbb{Q})$ vanishes.