Conjectured structure of quartic graphs with minimum spectral gap

Let $G$ be a connected quartic graph, meaning a connected $4$-regular graph, on $n\geq 11$ vertices. Let $M_1$ denote the specified middle block and $D_1,D_2,D_4,D_5$ the specified end blocks from the source. Choose non-negative integers $q$ and $r<5$ satisfying

$$n-11=5q+r.$$

A graph is formed by concatenating $q$ middle blocks and the indicated end blocks.

Quartic minimum-gap conjecture. The connected quartic graph on $n\geq 11$ vertices with minimum spectral gap is the unique graph $G$ described as follows: $G$ consists of $q$ middle blocks $M_1$, with end blocks chosen according to $r$. If $r=0$, both end blocks are $D_4$; if $r=1$, they are $D_4$ and $D_1$; if $r=2$, both are $D_1$; if $r=3$, they are $D_1$ and $D_2$; and if $r=4$, they are $D_4$ and $D_5$.

This conjecture seeks the precise extremal structure, beyond the paper's theorem describing the general path-like block structure of quartic graphs with minimum spectral gap. The block definitions are supplied by figures elsewhere in the paper, and the assertion is not marked as resolved in the provided text.