Ji's multiplier-spectrum injectivity conjecture

Let $f$ and $g$ be rational maps of degree $d\geq 2$ whose conjugacy classes are different. Let $\tau_d$ be the multiplier-spectrum map, and call two rational maps elementarily equivalent in the sense used in the source. A Lattès map is a rational map belonging to the Lattès class.

Multiplier-spectrum injectivity conjecture. If $\tau_d(f)=\tau_d(g)$, then either

1. $f$ and $g$ are Lattès maps; or
2. $f$ is elementarily equivalent to $g$.

This conjecture asserts that Lattès maps and elementary equivalence are the only obstructions to injectivity of the multiplier spectrum. It is attributed in the passage to Ji and the author, following a question of Pakovich; no resolution is supplied.