Failure of tensor-product compatibility for supports in Markov categories

Let $\mathsf{C}$ be a Markov category with supports, and let $p$ and $q$ be morphisms in $\mathsf{C}$. Write $S_f$ for the support object of a morphism $f$. Support tensor-product failure conjecture. There exists such a category and such morphisms for which

$$S_{p \otimes q} \not\cong S_p \otimes S_q.$$

The question concerns whether supports interact well with the monoidal structure. The source states that this remains open, while suggesting that the displayed compatibility need not hold.