The higher cs lower bound equality conjecture for simplicial polytopes

Let $P$ be a centrally symmetric simplicial $d$-polytope. The higher cs lower bound conjecture. If, for some $3\leq r<\left\lfloor\frac d2\right\rfloor$,

$$g_r(P)=g_r(\mathcal C^*_d),$$

then

$$g_{r+1}(P)=g_{r+1}(\mathcal C^*_d).$$

Here $\mathcal C^*_d$ is the $d$-dimensional cross-polytope. The cases $r=1$ and $r=2$ are known according to the source, while the stated higher cases remain open.