Relation between the modular categories SU(2)_n and Sp(n)_1 The online database [1] provides a list of some modular tensor categories classified by rank. Let us consider the two modular categories denoted kmA1_$\ell$ and kmC$\ell$_1 (i.e. Kac Moody $A_1$ level $\ell$ and $C_{\ell}$ level 1, respectively) appearing in this list at rank $\ell+1$ (with $3\le \ell \le 24$). I guess that these categories are the ones also called $SU(2)_{\ell}$ and $Sp(\ell)_1$ in the literature.
According to [1], these two modular categories have same type (i.e. list of qdims), at least between rank $4$ and $25$.
Question: Are they Grothendieck equivalent (i.e. same fusion ring up to isomorphism)?
[If not, what are the fusion rules for $Sp(\ell)_1$?]
Remark 1: That should be true for $\ell=4$ by Theorem 4.1 in [2], by some dimension argument.
Remark 2: They should not be equivalent as categories because according to [1], their central charge are different.

References
[1] Terry Gannon, Gerald Höhn, Hiroshi Yamauchi,...; Vertex Operator Algebras and Modular Categories; https://www.math.ksu.edu/~gerald/voas/mtc/mtc-byrank.html
[2] Paul Bruillard, Siu-Hung; Ng, Eric C. Rowell, Zhenghan Wang; On classification of modular categories by rank. Int. Math. Res. Not. IMRN 2016, no. 24, 7546–7588.
 A: Below is an email of Andrew Schopieray answering positively this question.

I randomly noticed a question of yours on mathoverflow about the fusion rules of some quantum group categories. Feel free to answer your own question based on the following.
The equivalence of the fusion rules you asked about is the most degenerate case of "rank-level duality" in type C, I believe (pre-1990's Lie theory results).   For future reference, Terry classified all the isomorphisms between the quantum group categories in the 90's as well: The automorphisms of affine fusion rings. Your isomorphism is contained in the last result of the paper.

It is also known that if you have the fusion rules of sl(2) (in general, as a tensor category), then you are equivalent to the semisimplification of U_q(sl(2)) at some root of unity q.  This is a result of Kazhdan-Wenzl from the 90's.  I think it's the one that's hard to access, called Reconstructing monoidal categories.  This paper cites it: Reconstructing Braided Subcategories of $SU(N)_k$ which asks similar questions.   The choice of root of unity for the parameter q explains why your central charge is different for the two categories.
