Let $$(\ast) \qquad g(\alpha,\beta,\gamma) := \sum_{\sigma,\omega,\pi} \text{sign}(\sigma\omega\pi) \, C(\alpha-\sigma,\beta-\omega,\gamma-\pi), $$ where $\alpha,\beta,\gamma \vdash n$ are partitions with $\le \ell$ rows viewed as vectors in $\Bbb R^\ell$, $C(x,y,z)$ is the number of $\ell\times \ell \times \ell$ contingency arrays with 2-dim sums given by vectors $x,y,z$, and the sum is over triples $\sigma,\omega,\pi$ of permutations of $\{0,1,\ldots,\ell-1\}$.

Now $g(\alpha,\beta,\gamma)$ is the Kronecker coefficient, and we have the inequality $$ g(\alpha,\beta,\gamma) \ge 0. $$ Finding a combinatorial proof of this inequality is a major problem in this case.

P.S. For $(\ast)$, see e.g. eq. (8) in our paper On the complexity of computing Kronecker coefficients.