Outage Probability Conjecture for MIMO channels

Consider a single-user MIMO Gaussian channel with $t$ transmitting antennas and $r$ receiving antennas, channel matrix ${\mathbf{H}}\in\mathbb{C}^{r\times t}$, and covariance matrices ${\mathbf{Q}}\geq 0$ satisfying $\operatorname{tr}({\mathbf{Q}})\leq P$. Let $P_{\mathrm{out}}(R,P)$ be the infimum of the outage probability over such covariance matrices. The matrices ${\mathbf{Q}}$ attaining this infimum should have an eigenvalue decomposition

$${\mathbf{Q}}={\mathbf{U}}{\mathbf{D}}{\mathbf{U}}^*,$$

where ${\mathbf{U}}$ is $t\times t$ unitary and, for some integer $1\leq k\leq t$,

$${\mathbf{D}}=\frac{P}{k}\operatorname{diag}(\underbrace{1,\dots,1}_{k},\underbrace{0,\dots,0}_{t-k}).$$

Outage Probability Conjecture. Every covariance matrix attaining the infimum has the stated form: power is allocated equally among some $k$ transmit dimensions and is zero on the remaining $t-k$ dimensions. The conjecture expresses the symmetry between the transmitters. It was generally accepted as plausible in the source, but the supplied text gives no resolution for the general MIMO case.