For simple, undirected graphs $G, H$, let $G\times H$ denote the [categorical product, or tensor product](https://en.wikipedia.org/wiki/Tensor_product_of_graphs), of $G$ and $H$. 

Let us call a  graph $G = (V,E)$ *(product-)reducible* if there are graphs $G_i = (V_i, E_i)$  with $|V_i| > 1$ for $i = 1,2$ and $G \cong G_1 \times G_2$. We call $G=(V,E)$ *properly irreducible* if $|V|> 1$ and $G$ is not reducible.

**Question.** Are there properly irreducible graphs  $G_i, H_i$ (possibly infinite) for $i=1,2$ such that $G_1\times G_2 \cong H_1\times H_2$, but $G_i \not\cong H_j$ for all $i,j\in\{1,2\}$?