Costa's corner-point conjecture for two-user Gaussian interference channels

Let $(R_1,R_2)$ be a rate pair in the capacity region of a two-user Gaussian interference channel with arbitrary positive cross-link gains $a_{12}$ and $a_{21}$ and power constraints $P_1$ and $P_2$. Define the single-user additive white Gaussian noise capacities

$$C_1=\frac{1}{2}\log(1+P_1),\qquad C_2=\frac{1}{2}\log(1+P_2).$$

Also define

$$R_1^*=\frac{1}{2}\log\left(1+\frac{a_{21}P_1}{1+P_2}\right),\qquad R_2^*=\frac{1}{2}\log\left(1+\frac{a_{12}P_2}{1+P_1}\right).$$

Costa's corner-point conjecture. For reliable communication at both receivers, the following should hold: if $R_2\geq C_2-\varepsilon$ for an arbitrary $\varepsilon>0$, then

$$R_1\leq R_1^*+\delta_1(\varepsilon),$$

where $\delta_1(\varepsilon)\to 0$ as $\varepsilon\to 0$; and if $R_1\geq C_1-\varepsilon$, then

$$R_2\leq R_2^*+\delta_2(\varepsilon),$$

where $\delta_2(\varepsilon)\to 0$ as $\varepsilon\to 0$. The conjecture addresses the two unknown corner points of the capacity region for weak Gaussian interference channels after a gap was identified in the proof of Costa's earlier result; its validity for arbitrary positive cross-link gains remains unresolved in the supplied source.