The finite Friedman–Stanley jump spectrum conjecture

Let $=^{+\alpha}$ denote the $\alpha$-fold Friedman–Stanley jump of equality, and let the spectrum of the meager ideal be the collection of Borel equivalence relations represented by homomorphisms into the relevant generic orbit structure modulo meager sets.

Spectrum conjecture. For each countable ordinal $\alpha$, $=^{+\alpha}$ is in the spectrum of the meager ideal.

This asserts that every countable finite or transfinite Friedman–Stanley jump occurs in the spectrum of the meager ideal. The supplied text does not state whether the conjecture is known or open.