The Gallai-Ramsey conjecture for books

Let $B_m$ be the book with $m$ pages, defined by

$$B_m=K_2+\overline{K_m}.$$

For a graph $H$, let $R(H,H)$ denote its two-color Ramsey number, and let $gr_k(K_3:B_m)$ be the minimum order of a complete graph whose every $k$-coloring contains either a rainbow $K_3$ or a monochromatic $B_m$.

Gallai-Ramsey conjecture for books. For $k\geq 2$,

$$gr_k(K_3:B_m)=\begin{cases}(R(B_m,B_m)-1)\cdot5^{(k-2)/2}+1 & \text{if $k$ is even,}\\2\cdot(R(B_m,B_m)-1)\cdot5^{(k-3)/2}+1 & \text{if $k$ is odd.}\end{cases}$$

The conjecture asserts that the lower bounds obtained in the paper are sharp, giving exact Gallai-Ramsey numbers for books. The surrounding conclusion presents only bounds and says that the conjecture is being offered because the authors believe the lower bound to be sharp; its status is not resolved in the supplied source.