The proper hat-guessing number conjecture for two-spine book graphs

Let $B_{2,n}$ denote the two-spine book graph with $n$ pages, and let $\operatorname{HGP}(G)$ be the proper hat-guessing number of a graph $G$. Two-spine book graph conjecture. For every $n\geq4$,

$$\operatorname{HGP}(B_{2,n})=\min\{n+3,11\}.$$

The preceding results establish the stated values in several ranges and show stabilization at $11$ for all $n\geq n_0$. The lower bound remains open for the finite range before stabilization.