Felsner–Weil connectivity conjecture for flip graphs of signotopes

An $r$-signotope is a mapping from the $r$-subsets of an $n$-element set to $\\{+,-\\}$ satisfying the relevant monotonicity condition, and a flip changes the sign of a single $r$-set. Let $\mathbf{G}^r_n$ be the flip graph of $r$-signotopes on $n$ elements. Felsner–Weil conjecture. For every $r\ge 3$, the flip graph $\mathbf{G}^r_n$ has minimum degree $n-r+1$ and is $(n-r+1)$-connected. The cases $r=1$ and $r=2$ are known, corresponding respectively to hypercubes and permutahedra; the assertion for higher ranks remains unresolved.