Cycle obstruction conjecture for ground-graph-independent blowup Ramsey numbers

Let $r\in\mathbb{N}$ and let $H$ be a graph containing a cycle. Write $G\stackrel{r}{\to}H$ when every $r$-colouring of the edges of $G$ contains a monochromatic copy of $H$, and let $H[t]$ denote the $t$-blowup of $H$.

Cycle obstruction conjecture. There exists an integer $t$ such that, for every $s$, there is a graph $G$ satisfying

$$G\stackrel{r}{\to}H\quad\text{but}\quad G[s]\not\stackrel{r}{\to}H[t].$$

Together with the proved ground-graph-independent behaviour for forests, this conjecture would classify forests as exactly the finite graphs whose blowup Ramsey numbers do not depend on the ground graph. The source establishes the required obstruction for 3-chromatically connected graphs, while the assertion for every graph containing a cycle remains open.