Path-pairable Cartesian products of two non-path-pairable graphs

Let $G$ and $H$ be graphs, and let $G\square H$ denote their Cartesian product. A graph is path-pairable if every pairing of distinct vertices can be joined by pairwise edge-disjoint paths.

Cartesian-product conjecture. There exist non-path-pairable graphs $G$ and $H$ such that

$$G\square H$$

is path-pairable.

The Cartesian product of two path-pairable graphs need not be path-pairable, so this conjecture asks whether path-pairability of either factor is necessary at all for path-pairability of the product. The source describes this as an open question and states that the proposed condition is believed not to be necessary.