Higher Reidemeister trace characterization conjecture

Let $M$ be a smooth closed manifold, let $\mathcal{H}(M)$ be its stable $h$-cobordism space, and suppose the higher Reidemeister trace constructions define a map

$$RT:\mathcal{H}(M)\to\Omega^{\infty+1}\Sigma^{\infty}\frac{\mathcal{L}M}{M}.$$

Let $\Xi_{{{l}}}^B$ and $\Xi_{{{r}}}^B$ be the parametrised-spectrum operations associated with a smooth fiber bundle over $B$, and let $\mu_{{{l}}}^{M\times B}$ and $\mu_{{{r}}}^{M\times B}$ denote the corresponding multiplication operations. Higher Reidemeister trace characterization conjecture. There are homotopies of maps of parametrised spectra

$$\Xi_{{{l}}}^B\simeq\mu_{{{l}}}^{M\times B}(\cdot\times[M],[RT_{diag}]) \quad\text{and}\quad \Xi_{{{r}}}^B\simeq\mu_{{{r}}}^{M\times B}([\overline{RT}_{diag}],[M]\times\cdot).$$

The conjecture identifies the two parametrised obstruction operations with multiplication by higher Reidemeister traces. The source does not state that this characterization is known, so its status is open.