Let $(N \subset M)$ be an irreducible finite index depth $n$ subfactor. Let $P = P(N \subset M)$ its planar algebra.

Let $(B_i)$ be the finite sequence of $N$-$N$-bimodules appearing in the principal graph.

Let $2m = n$ if $n$ even, else $2m=n+1$.

Let $p_i \in P_{2m,+}$ be the minimal central projection related to the $N$-$N$-bimodule $B_i$.

*Question*: Is there a planar tangle $T: P_{2m,+} \otimes P_{2m,+} \to P_{2m,+}$ such that $T(p_i \otimes p_j) = \sum_{k} n_{ij}^k p_k $ with $B_i \boxtimes B_j = \bigoplus_k M_{ij}^k \otimes B_k$ and $dim(M_{ij}^k)= n_{ij}^k$ (the fusion coefficients)?

Else, is there such a $T$ if we only consider the range support? the central support?

*Remark:* If $n = 2$, such a $T$ exists, it's the coproduct (see here).

Then, a generalization of the coproduct on $P_{2m,+}$ could do the job.