Suppose I have a "character table" $\Lambda = (\lambda_{ij})$ of a purported tensor category $\mathcal{T}$ (perhaps some adjectives are needed); right now $\Lambda$ is just an $n \times n$ integer matrix with properties which I shall now list. As in the case of a finite group, we'll assume that the columns of $\Lambda$ are orthogonal and that for each index $1 \leq i \leq n$ there exists a positive rational value ${\frak{z}}(i)$ with $\sum_i {\frak{z}}(i) = 1$ such that
\begin{equation} \sum_l \lambda_{i,l} \, \lambda_{j,l} \, {\frak{z}}(l) = \delta_{i,j} \end{equation}
We require that the entries of the first column $\lambda_{i,1}$ are strictly positive; they correspond to "dimensions" of the simple objects in our purported tensor category $\mathcal{T}$. Furthermore let's assume that for any triple of rows $i, j, k$ we have non-negative integer Kronecker coefficients $g_{i,j}^k$ defined in the usual way by:
\begin{equation} g_{i,j}^k \, := \, \sum_l \lambda_{i,l} \, \lambda_{j,l} \, \lambda_{k,l} \, {\frak{z}}(l) \end{equation}
Because $\Lambda$ is an integer matrix each Kronecker coefficient $g_{i,j}^k$ will be fully symmetric with respect to its three indices. Let's require that $g_{i,j}^1 =\delta_{i,j}$ (which forces $g_{1,i}^j = g_{i,1}^j =\delta_{i,j}$ by symmetry). As described these Kroncker coefficients are structure constants for an n-dimensional, commutative, unital algebra $\Bbb{A}$: It has basis vectors $\Bbb{a}_1, \dots, \Bbb{a}_n$ with unit $1 = \Bbb{a}_1$ and product given by
\begin{equation} \Bbb{a}_i \cdot \Bbb{a}_j \, = \, \sum_k g_{i,j}^k \, \Bbb{a}_k \end{equation}
On top of all this, let's insist that $\Bbb{A}$ is associative (in case that doesn't come for free).
Question: Given the circumstances which I've described, can it happen that Sebastien Palcoux's non-negativity condition for column triples of $\Lambda$ is violated (see https://mathoverflow.net/questions/344968/a-new-combinatorial-property-for-the-character-table-of-a-finite-group); namely there exist columns $i, j, k$ such that
\begin{equation} \sum_l { {\lambda_{l,i} \, \lambda_{l,j} \, \lambda_{l,k}} \over {\lambda_{l,1}} } < 0 \end{equation}
If this is possible, how does the failure of his non-negativity property restrict any purported tensor category $\mathcal{T}$ whose Grothendieck ring might coincide with $\Bbb{A}$? E.g. can we conclude that $\mathcal{T}$ can't be a fusion category ? If not, is there some weakening of the notion of fusion category which is appropriate ?
p.s. I'm sorry if the question is a bit vague; I'm still trying to get a lay of the land in the realm of Fusionism.