Joint empirical density formula for left and right eigenvectors

Let $X$ be an $N\times N$ matrix, let $z$ be an eigenvalue, and let ${\bf v}$ and ${\bf u}$ be corresponding right and left eigenvectors. Normalize the right eigenvector by ${\bf v}^*{\bf v}=1$ and impose ${\bf v}^*{\bf u}=1$. With respect to the measure $d^2z\,d^{2N}{\bf u}\,d\mu_H({\bf v})$, where $d\mu_H$ is the relevant Haar measure on normalized right eigenvectors, let $\Pi_N(z,\mathbf{v},\mathbf{u})$ denote their joint empirical density. Left-right eigenvector density conjecture. The density is given by

$$\Pi_N(z,\mathbf{v},\mathbf{u})=K_N\left|{\,\rm \det}\:\left(X-z{\bf 1}_N\right)\right|^2\left|\frac{d}{dz}{\,\rm \det}\:\left(X-z{\bf 1}_N\right)\right|^2$$ $$\times\delta^{(2N)}\left(\left(X-z{\bf 1}_N\right)\mathbf{v}\right)\,\delta^{(2N)}\left(\left(X-z{\bf 1}_N\right)^*\mathbf{u}\right)\delta^{(2)}({\bf v}^*\mathbf{u}-1),$$

where $K_N$ is the appropriate normalization constant. A formula of this kind would extend the Kac–Rice approach from right-eigenvector statistics to joint left- and right-eigenvector statistics and could provide access to overlap observables, but the source states that this direction is not yet complete.