The dual Borel-pairing conjecture for Morita classes

Assume $(p_0)_*(\mu_k)=0$, and construct a cycle $u\in Z_{4k+1}(\mathrm{GL}(2k+3,\mathbb{Z});\mathbb{Q})$ by bounding the relevant abelian cycle in $\mathrm{Out}\,F_{2k+3}$ and its image in $\mathrm{GL}(2k+2,\mathbb{Z})$. Let $\beta_{2k+1}$ denote the Borel regulator class in degree $4k+1$. Dual Borel-pairing conjecture. For a suitable choice of $u$,

$$\langle\beta_{2k+1},[u]\rangle\neq 0.$$

If true, this would imply that the Morita class $\mu_k$ is nonzero; the source discusses a possible verification for $k=1$, but the conjecture is otherwise open.