DPC optimality conjecture for the K-user quasi-static fading broadcast channel

Consider the $K$-user quasi-static scalar fading broadcast channel without transmit CSI,

$$Y_k = \sqrt{A_k}X + Z_k,\quad k=1,\ldots,K,$$

where $Z_k \sim \mathcal{CN}(0,1)$), $X$ has average power constraint $\rho$, each $A_k$ is constant over the coding block and known only at decoder $k$, and the $A_k$ are mutually independent. Let $G_k(\epsilon_k)$ denote the corresponding $\epsilon_k$-outage channel-gain quantile, and let $\mathcal{R}^\mathrm{dpc}(\rho,\underline{\epsilon})$ be the DPC outage achievable rate region defined by the power-allocation parameters $\gamma_k\geq 0$ with $\sum_{k=1}^K\gamma_k=1$.

DPC optimality conjecture. If the $K$ decoders are ordered such that

$$G_1(\epsilon_1) \geq G_2(\epsilon_2) \geq \cdots \geq G_K(\epsilon_K),$$

then $\mathcal{R}^\mathrm{dpc}(\rho,\underline{\epsilon})$ is the boundary of the $\underline{\epsilon}$-outage capacity region $\mathcal{C}^\mathrm{out}(\rho,\underline{\epsilon})$.

The conjecture extends the established two-user comparison between DPC and time division to the general $K$-user quasi-static fading broadcast channel. Its motivation is that, after replacing each fading gain by its outage quantile and ordering the decoders by these gains, the DPC region coincides with the capacity region of the corresponding degraded broadcast channel without fading. The supplied text gives no resolution, so the conjecture is recorded as open.