Length-spectrum versus multiplier-spectrum conjecture for non-Lattès maps

Let $d\geq 2$, and let $f,g$ be non-Lattès rational maps of degree $d$. Let $\tau_d$ denote the multiplier spectrum map, and let $\overline{g}$ be the coefficientwise complex conjugate of $g$. Length-spectrum conjecture. If $f$ and $g$ have the same length spectrum, then

$$\tau_d(f)=\tau_d(g)\quad\text{or}\quad\tau_d(f)=\tau_d(\overline{g}).$$

This gives a more precise proposed description of pairs of rational maps with the same length spectrum, strengthening the preceding generic statement to all non-Lattès maps. Its resolution is not supplied in the source.