General bipartite graph conjecture for Gallai-Ramsey numbers

Let $H$ be a connected bipartite graph, let $R(H,H)=R$ be its two-color Ramsey number, and let $s(H)$ denote the parameter used in the source. The general bipartite Gallai-Ramsey conjecture.

$$gr_k(K_3:H)=R+(s(H)-1)(k-2).$$

This proposes a broad exact formula for Gallai-Ramsey numbers of connected bipartite graphs, extending the paper's bounds and examples. The source does not specify the definition of $s(H)$ or report a resolution.