Uniqueness conjecture for minimum-spectral-gap quartic graphs

For each $n\ge 11$, let $\mathcal{G}_n$ be the quartic graph of order $n$ constructed from the blocks displayed in the paper, and call a quartic graph minimal when it has minimum spectral gap among quartic graphs of order $n$. Uniqueness conjecture. For any $n\ge 11$, the graph $\mathcal{G}_n$ is the unique minimal quartic graph of order $n$.

This is a structural claim identifying the extremal quartic graph, complementing the paper's asymptotic determination of the minimum spectral gap for connected quartic graphs. The supplied text does not state whether the uniqueness claim has been proved beyond its conjectural presentation.