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.


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