The strong form of Oliver's conjecture for the subgroup Y(S)

Let $p$ be a prime and $S$ a finite $p$-group. Let $\mathcal{Y}(S)$ be the characteristic subgroup introduced in the source; for odd $p$ it satisfies $\mathcal{Y}(S)\leq\mathfrak{X}(S)$, while for $p=2$ one has $\mathcal{Y}(S)=S$. Let $J(S)$ denote the Thompson subgroup of $S$.

Strong form of Oliver's conjecture.

$$J(S)\leq\mathcal{Y}(S).$$

This strengthens Oliver's conjecture: it is automatic for $p=2$, and for odd primes it implies the original conjecture. The paper introduces it as a conjecture and reports verification for the stated families in its abstract, not a general resolution.