Extremal triangle-count conjecture for bounded book number

Let $G$ be a graph on $n$ vertices, and let $b(G)$ be its book number. For nonnegative integers $b$ and $n$ with $b\leq n/4$, let $S_{b,n}$ be the graph obtained by blowing up the $3$-prism so that four parts have size $b$ and the other two have sizes $\lfloor(n-4b)/2\rfloor$ and $\lceil(n-4b)/2\rceil$. Thus $S_{b,n}$ has $\lfloor n^2/4\rfloor$ edges, book number $b$ when $b\geq n/6$, and $b^2(n-4b)$ triangles. Extremal triangle-count conjecture. If $n/6\leq b<n/4$, then every graph on $n$ vertices with at least $\lfloor n^2/4\rfloor$ edges and book number at most $b$, other than the balanced complete bipartite graph, has at least

$$b^2(n-4b)$$

triangles, with equality if and only if the graph is $S_{b,n}$. This conjecture predicts the exact minimum triangle count in the stated range and is confirmed in the source for $b/n=1/6$ and for $0.2495\leq b/n<1/4$.