# Existence of a unitary fusion category with this relation ruled out on finite groups

In this answer, Geoff ruled out the existence of a finite group $$G$$ such that the fusion category $$\mathrm{Rep}(G)$$ has simple objects $$5_1$$ and $$7_1$$ of FPdim $$5$$ and $$7$$ resp., with (for some object $$X$$): $$5_1 \otimes 5_1 = 1 \oplus 5_1 \oplus 7_1 \oplus X$$ and in this comment he thinks that it is also probably true if $$(5,7)$$ is replaced by any twin prime pair $$(p,q)$$ with $$p>3$$. This leads to ask about an extension to (unitary) fusion category:

Question: Is there a (unitary) fusion category having two simple objects $$p_1$$ and $$q_1$$ of FPdim $$p$$ and $$q$$ resp., with $$(p,q)$$ a twin prime pair, $$p>3$$ and (for some object $$X$$) satisfying the following? $$p_1 \otimes p_1 = 1 \oplus p_1 \oplus q_1 \oplus X$$

Specifically interested in $$(p,q) = (5,7)$$ and $$X = 5_2 \oplus 7_2$$, so if one such example is known (or cannot exist), a proof or a reference would be welcome in answer.