Ji's length-spectrum rigidity conjecture for rational maps

Let $f$ and $g$ be non-Lattès rational maps of degree $d\geq 2$. For each $n\geq 1$, let $L_n(f)$ denote the unordered collection of absolute derivatives at the fixed points of $f^n$, and let the length spectrum be the sequence $(L_n(f))_{n\geq 1}$. Let $\tau_d(f)$ denote the multiplier-spectrum invariant introduced in the source, and let $\overline g$ be the complex conjugate of $g$.

Length-spectrum rigidity conjecture. If $f$ and $g$ have the same length spectrum, then

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

The conjecture gives a precise description of rational maps with identical length spectra. It is presented as a conjecture proposed by Ji and the author; no resolution is supplied in the passage.