Higher Whitehead torsion and Reidemeister trace compatibility conjecture

Let $M$ be a smooth closed manifold, let $\mathcal{H}(M)$ be its stable $h$-cobordism space, and consider the Dennis trace map $tr$ on $THH(\Sigma^{\infty}_+\Omega M)$, Waldhausen's splitting map, the higher Reidemeister trace $RT$, and the equivalence

$$THH(\Sigma^{\infty}_+\Omega M)\simeq\Sigma^{\infty}_+\mathcal{L}M.$$

Higher Whitehead torsion and Reidemeister trace compatibility conjecture. The following diagram commutes up to natural homotopy:

$$\begin{array}{ccc} \Omega\Omega^\infty K[\Sigma^\infty_+\Omega M]&\longrightarrow&\mathcal{H}(M)\\ \downarrow{\scriptstyle\Omega tr}&&\downarrow{\scriptstyle RT}\\ \Omega\Omega^\infty THH(\Sigma^\infty_+\Omega M)&\longrightarrow&\Omega\Omega^\infty\Sigma^\infty(\mathcal{L}M/M) \end{array}$$

where the top horizontal map is given by Waldhausen's splitting theorem and the bottom horizontal map is induced by the displayed equivalence. This conjecture relates higher Whitehead torsion, the Dennis trace, and higher Reidemeister traces. The source provides no resolution status, so the conjecture is recorded as open.