Braun–Bruegge facet bounds conjecture for symmetric edge polytopes

Let $G$ be a connected graph with $n\geq 3$ vertices. Let ${\mathcal P}_G$ be its symmetric edge polytope, and let $N({\mathcal P}_G)$ denote the number of its facets. A $1$-sum of graphs is formed by taking their union along a common vertex. Braun–Bruegge's conjecture. If $n$ is odd, then

$$3\cdot 2^{\frac{n-1}{2}}-2\leq N({\mathcal P}_G)\leq 6^{\frac{n-1}{2}}.$$

Moreover, equality on the left holds if and only if $G=K_{(n-1)/2,(n+1)/2}$, while equality on the right holds if and only if $G$ is the $1$-sum of $(n-1)/2$ triangles. If $n$ is even, then

$$2^{\frac{n}{2}+1}-2\leq N({\mathcal P}_G)\leq 14\cdot 6^{\frac{n}{2}-2}.$$

Moreover, equality on the left holds if and only if $G=K_{n/2,n/2}$, while equality on the right holds if and only if $G$ is the $1$-sum of $K_4$ with $n/2-2$ triangles.

The conjecture concerns sharp lower and upper bounds for facets of symmetric edge polytopes. The paper states that it proves the conjecture for every graph that is the join of two graphs, equivalently every connected graph whose complement is disconnected; the supplied context does not establish the conjecture in full generality.