Ji's generic length-spectrum injectivity conjecture

For every $d\geq 2$, let ${\mathcal M}_d$ be the moduli space of degree-$d$ rational maps, let $E_d\subset {\mathcal M}_d$ be a subset defined over $\mathbb R$, and let $\overline g$ denote the complex conjugate of $g$. Two rational maps are conjugate when they represent the same point of the moduli space.

Generic length-spectrum injectivity conjecture. There is a Zariski closed proper subset $E_d\subset {\mathcal M}_d$ defined over $\mathbb R$ such that, for every $[f]\notin E_d$, if some $g\in\operatorname{Rat}_d(\mathbb C)$ has the same length spectrum as $f$, then either $g$ or $\overline g$ is conjugate to $f$.

This is the length-spectrum analogue of the generic injectivity theorem for the multiplier spectrum. It is presented as a conjecture of Ji and the author, and the supplied passage gives no resolution.