A fusion ring is a finite dimensional complex space $\mathbb{C}\mathcal{G}$ together with a distinguished basis $\mathcal{G} = \{ h_1,...,h_r\}$ and fusion rules $ h_i \cdot h_j = \sum_k n_{ij}^kh_k $, with $n_{ij}^k \in \mathbb{N}_{\geq 0}$ satisfying:

• Neutral: $n_{1i}^j = n_{i1}^j = \delta_{ij}$
• Dual: $\forall i \ \exists!j $ (noted $i^*$) such that $n_{ij}^1>0$
• Associativity: $\sum_s n_{ij}^sn_{sk}^t = \sum_s n_{jk}^sn_{is}^t$
• Frobenius-Perron reciprocity: $n_{ij}^k = n_{i^*k}^j = n_{kj^*}^i$

Remark: $\mathbb{C}\mathcal{G}$ admits a structure of finite dimensional ${\rm C}^*$-algebra (take $h_i^* = h_{i^*}$).
Frobenius-Perron theorem: there is a unique $*$-homomorphism $d:\mathbb{C}\mathcal{G} \to \mathbb{C}$ with $d(\mathcal{G}) \subset (0,\infty)$.

The fusion ring $\mathbb{C}\mathcal{G}$ is called integral if every $d(h_i)$ is an integer. Its rank is the cardinal of $\mathcal{G}$, and its Frobenius-Perron dimension (FPdim) is $\sum d(h_i)^2$. It is simple if for any fusion subring $\mathbb{C}\mathcal{S} \subseteq \mathbb{C}\mathcal{G}$ with $\mathcal{S} \subseteq \mathcal{G}$, we have $\mathcal{S} = \{ h_1 \}$ or $\mathcal{G}$.

We have checked by SAGE (by using this code) that the only integral simple fusion ring of rank $\leq 5$ and FPdim $\leq 11500$ is the Grothendieck ring of the simple group $A_5$. It is of rank $5$ and FPdim $60$ (we don't consider the Grothendieck rings of prime order cyclic groups).

Question: Is there an integral simple fusion ring of rank $ \leq 5$ and FPdim $>60$?

Digression: a fusion ring is called non-trivial if it is not the Grothendieck ring of a finite group. The first non-trivial integral simple fusion ring found by SAGE is of rank $7$ and FPdim $210$ (see here).