The Lefschetz-to-$g$ hierarchy conjecture for homology spheres

Let $L$ be a homology $(d-1)$-sphere. Write $\binom{a}{b}$ only through the interval notation in the claim, where $[a,b]$ denotes the corresponding set of integers, and let

\Delta(L)$ be the relevant symmetric shifting of $L$. **Lefschetz-to-$g$ hierarchy conjecture.** The following assertions hold for $L$: (1) If $S\in\Delta(L)$, $|S|=k\leq\lfloor d/2\rfloor$ and $S\cap[d-k+1]=\emptyset$, then $S\cup[k+2,d-k+1]\in\Delta(L)$; this is equivalent to

\Delta(K)\subseteq\Delta(d)$and, in the symmetric case, to$L$being strong-Lefschetz. (2) If$S\in\Delta(L)$,$|S|=k<\lfloor d/2\rfloor$and$S\cap[d-k+1]=\emptyset$, then$S\cup[\lceil d/2\rceil+2,d-k+1]\in\Delta(L)$; in the symmetric case, this is equivalent to$L$being weakly weak-Lefschetz. (3)$g(L)$is an$M$-vector. The paper presents these assertions as a hierarchy in which each assertion implies the next; the source does not state that the hierarchy has been resolved.