Shaping-loss conjecture for the Gaussian doubly-dirty MAC

Let the Gaussian doubly-dirty multiple access channel have high signal-to-noise ratio and strong interference. Denote its sum capacity by $C^{\sum}$ and the sum rate of the best known single-letter characterization by $R_{BSL}^{\sum}$, as defined by the corresponding single-letter region. Shaping-loss conjecture. For the Gaussian doubly-dirty MAC, the best known single-letter expression has sum-rate loss

$$C^{\sum}-R_{BSL}^{\sum}=\frac{1}{2}\log_2\Big(\frac{\pi e}{6}\Big)\approx 0.254\;\text{bit}$$

with respect to the sum capacity. This conjecture concerns whether single-letter random-coding characterizations incur the familiar shaping loss; the paper provides a partial proof, while the optimization needed for a complete proof remains open.