Bessenrodt–Panova conjecture on self-conjugate partitions

Let $n$ be a positive integer, and let a partition be self-conjugate if it equals its conjugate. Its Durfee size is the side length of its largest square contained in its Young diagram. Bessenrodt–Panova conjecture. For every $n$, there exists $k(n)$ such that the tensor square of every self-conjugate partition of $n$ whose Durfee size is at least $k(n)$ and which is not the $k\times k$ partition satisfies the Tensor Square Conjecture. The source presents this as a conjecture about the shapes whose tensor squares contain every irreducible representation, and gives no resolution.