Higher-dimensional Balinski-type connectivity conjecture for cell complexes

Let ${\mathcal C}$ be a $d$-dimensional Cohen–Macaulay regular cell complex with the intersection property. For each $k$, let ${\mathcal G}_k({\mathcal C})$ be the simple graph whose vertices are the $k$-dimensional cells of ${\mathcal C}$, with two cells adjacent when they are contained in a common $(k+1)$-dimensional cell.

Higher-dimensional Balinski-type connectivity conjecture. The graph ${\mathcal G}_k({\mathcal C})$ is

$$(k+1)(d-k)$$

-connected if $0\leq k\leq d-3$, and is $d$-connected if $k=d-2$.

This statement generalizes Balinski's theorem and the cited theorem of Fløystad, which is the case $k=0$. The source presents it as a statement following the known $k=0$ case; no resolution is supplied here.