Durfee squares, symmetric partitions and bounds on Kronecker coefficients

We resolve two open problems on Kronecker coefficients g(λ,μ,ν) of the symmetric group. First, we prove that for partitions λ,μ,ν with fixed Durfee square size, grow at most polynomially. Second, show maximal g(λ,λ,λ) self-conjugate λ superexponentially. also give applications to explicit special cases.