Fox–Luo–Wigderson's dependence-on-the-ground-graph conjecture for blowup Ramsey numbers

Let $G$ and $H$ be graphs, and let $r,t\geq 2$. Write $G\stackrel{r}{\to}H$ when every $r$-colouring of the edges of $G$ contains a monochromatic copy of $H$. For $t\in\mathbb{N}$, let $H[t]$ be the $t$-blowup of $H$, and let $B(G\stackrel{r}{\to}H;t)$ be the minimum $n$ such that every $r$-colouring of the edges of $G[n]$ contains a monochromatic canonical copy of $H[t]$.

Fox–Luo–Wigderson's conjecture. There exists a graph $H$ and integers $r,t\geq 2$ such that there are graphs $G_1,G_2,\dots$ with $G_i\stackrel{r}{\to}H$ for every $i$ and

$$\sup_i B(G_i\stackrel{r}{\to}H;t)=\infty.$$

This asserts that the dependence on the ground graph $G$ in the upper bound for blowup Ramsey numbers is necessary for some graphs $H$. The source gives the more specific prediction that this holds for a triangle with $r=t=2$.