Connectivity conjecture for edge slide graphs of hypercube signatures

Let $Q_n$ be the $n$-dimensional hypercube, let $\mathcal{S}=(a_1,\dots,a_n)$ be a signature of $Q_n$, and let ${\mathcal{E}(\mathcal{S})}$ be the edge slide graph induced by the spanning trees with signature $\mathcal{S}$. A signature is called irreducible, quasi-irreducible, or strictly reducible according to the classification defined in the paper. Connectivity conjecture. The graph ${\mathcal{E}(\mathcal{S})}$ is connected if and only if $\mathcal{S}$ is irreducible or quasi-irreducible. The theorem proved in the paper shows that every strictly reducible signature is disconnected, so the conjecture asserts that strict reducibility is the only obstruction to connectivity. Whether the edge slide graph is connected for every irreducible or quasi-irreducible signature remains open.