Lakshminarayan–Puchala–Życzkowski conjecture for moments of squared unimodular random matrices

For a positive integer $N$, let $U_N$ be the random $N\times N$ matrix with independent entries uniformly distributed on the unit circle, let $U_N^*$ be its adjoint, and define

$$\rho_N=\frac{1}{N^2}U_NU_N^*.$$

Let $\operatorname{tr}$ denote the normalized trace, and for positive integers $k$ and $N$ define

$$f_{k,j}:=\frac{1}{k+1}\binom{2k+2}{k-j}\binom{k+j}{j}.$$

Lakshminarayan–Puchala–Życzkowski conjecture. For $k$ and $N$ positive integers, it holds that

$$\mathbb{E}[\operatorname{tr}(\rho_N^k)]=N^{-2k-1}\sum_{j=2}^{k+1}(-1)^{k-j+1}f_{k-1,k-j+1}N^j,$$

where the $f_{k,j}$ are the elements of the Borel triangle.

The conjecture proposed a formula for the moments of the mean empirical spectral distribution of the squared unimodular random matrix. The paper states that its result disproves this formula.