The universal k-equivalence descent conjecture for the pair of pants

Let $M$ be the pair of pants, and let $k$-equivalence mean equality of geometric intersection numbers with every class of curves having self-intersection number $k$.

Universal k-equivalence descent conjecture. On the pair of pants, $k$-equivalence implies $(k-1)$-equivalence for all positive integers $k$.

This is the strongest formulation proposed in the source and contains the earlier conjecture for $k\geq 2$ as a special case. The paper proves descent to $1$- and $2$-equivalence in the relevant range, while the assertion for all positive integers remains open in the source.