Gamma-zero characterization conjecture for k-connected graph symmetric edge polytopes

For $k\geq 2$, let $G_{n,k}$ be the graph obtained from $K_{k,n-k}$ by adding all edges among the $k$ vertices in the first part; equivalently, it is the $k$-fold cone over $n-k$ isolated vertices. Let $\mathcal{P}_G$ be the symmetric edge polytope of a graph $G$ on $n$ vertices, and let $\gamma_k(\mathcal{P}_G)$ denote its $k$th gamma coefficient. A graph is $k$-connected if deleting fewer than $k$ vertices leaves it connected.

Gamma-zero characterization conjecture. If $k\in\mathbb{N}$ and $G$ is a $k$-connected graph on $n$ vertices, then

$$\gamma_k(\mathcal{P}_G)=0$$

if and only if $n<2k+1$, or $n\geq 2k+1$ and

$$K_{k,n-k}\subseteq G\subseteq G_{n,k}.$$

The conjecture generalizes the stated characterization for $\gamma_2=0$ and has been verified computationally for small values of $k$ and $n$. Its general status remains open.