Tangled labelings versus quasi-tangled labelings for inflated rooted trees

Let $P$ be an inflated rooted tree poset with deflated leaves. A labeling is tangled if its sorting time is $n-1$, and quasi-tangled if its sorting time is $n-2$.

Tangled–quasi-tangled inequality. The number of tangled labelings of $P$ is less than or equal to the number of quasi-tangled labelings of $P$.

The conjecture compares two successive sorting-time classes for inflated rooted trees with deflated leaves; no resolution is supplied, so it remains open.