Conjecture on optimal beamforming vectors for the two-user broadcast channel

Let ${\bf\Sigma}_1$ and ${\bf\Sigma}_2$ denote the channel covariance matrices, let $\rho$ be the signal-to-noise ratio, and let $\alpha^{\star}(\rho)$ and $\beta^{\star}(\rho)$ maximize $E[R_1]+E[R_2]$ subject to the candidate beamforming vectors ${\bf w}_{1,\,\sf cand}(\rho)$ and ${\bf w}_{2,\,\sf cand}(\rho)$. Write ${\sf Dom.\ eig}(\bullet)$ for the unit-norm dominant eigenvector operation. Optimal beamforming conjecture. In the intermediate-$\mathsf{SNR}$ regime, the ergodic sum-rate is maximized by

$${\bf w}_{1,\,\sf opt}=e^{j\nu_1}{\sf Dom.\ eig}\left(\left(\alpha^{\star}(\rho){\bf\Sigma}_2+{\bf I}\right)^{-1}{\bf\Sigma}_1\right),$$ $${\bf w}_{2,\,\sf opt}=e^{j\nu_2}{\sf Dom.\ eig}\left(\left(\beta^{\star}(\rho){\bf\Sigma}_1+{\bf I}\right)^{-1}{\bf\Sigma}_2\right),$$

for some $\nu_i\in[0,2\pi)$, $i=1,2$. The claim proposes the optimality structure of the paper's beamforming scheme in the intermediate-SNR regime; the authors state that they are unable to prove it and motivate it by numerical studies.