Ji's small-period multiplier-spectrum injectivity conjecture

For $d\geq 2$, let ${\mathcal M}_d(\mathbb C)$ be the moduli space of degree-$d$ rational maps, and let $N_1$ and $N_2$ be the numbers of fixed points of the first and second iterates counted in the source. Let $\tau_{d,2}$ be the multiplier-spectrum morphism using periods at most $2$.

Small-period multiplier-spectrum injectivity conjecture. For every $d\geq 2$, the morphism

$$\tau_{d,2}:{\mathcal M}_d(\mathbb C)\to \mathbb C^{N_1}/\Sigma_{N_1}\times \mathbb C^{N_2}/\Sigma_{N_2}$$

is generically injective.

The conjecture would establish generic injectivity using only multipliers from the first two periods, avoiding the potentially very large period bound in the full multiplier spectrum. The supplied passage presents it as an open conjecture of Ji and the author.