Fox, He and Wigderson's Ramsey goodness conjecture for complete multipartite graphs and books

Let $k,p,t\geq 2$ be integers. For a complete $p$-partite graph $H=K_p(a_1,\dots,a_p)$ with $1\leq a_1\leq\cdots\leq a_{p-1}\leq t$ and $a_p\leq\delta n$, let $B_{k,n}$ be the book graph on $n$ vertices consisting of $n-k$ copies of $K_{k+1}$ sharing a common $K_k$. Define $d_k(n,K_{a_1,a_2})$ as the maximum $d$ for which there is an $(n+d-1)$-vertex $K_{a_1,a_2}$-free graph in which at most $k-1$ vertices have degree less than $d$. Fox, He and Wigderson's conjecture. For every $k,p,t\geq 2$, there exists $\delta>0$ such that for all $n\geq 1$, whenever $1\leq a_1\leq\cdots\leq a_{p-1}\leq t$ and $a_p\leq\delta n$,

$$r(K_p(a_1,\dots,a_p),B_{k,n})=(p-1)(n-1)+d_k(n,K_{a_1,a_2})+1.$$

This extends known Ramsey-goodness results for books without using the regularity lemma; the conjecture is stated as open in the source, although the paper notes that it fails in many cases and remains open for $a_1\geq2$.