Extremal blowup conjecture for the maximum spectral gap $s_{1,j}$

Let $j\geq 1$. For a graph $G$, write $s_{1,j}$ for the maximum of the normalized spectral gap under consideration in the paper, and let $K_n^*$ denote the complete graph $K_n$ with loops on all vertices. The graph $K_{j+2}\cup K_{j+1}^*$ is the disjoint union of $K_{j+2}$ and $K_{j+1}^*$; a $t$-blowup replaces every vertex by an independent set of size $t$ and every edge or loop according to the corresponding adjacency pattern.

Extremal blowup conjecture. For any $j\geq 1$, we have

$$s_{1,j}=\frac{j+2}{2j+3}.$$

If the conjecture holds, the lower bound is achieved by any graph which is a $t$-blowup of $K_{j+2}\cup K_{j+1}^*$.

The construction gives the lower bound $s_{1,j}\geq (j+2)/(2j+3)$, while the paper notes that this value is close to the known upper bound for large $j$. The asserted equality and the extremal characterization remain open in the supplied source.