Explicit inner and outer surfaces for Gaussian pattern recognition

Let $X$ and $Y$ be zero-mean Gaussian random variables with correlation coefficient $\rho_{xy}$. For correlation coefficients $\rho_{xu}$ and $\rho_{yv}$ determined by

$$r_x=-\frac{1}{2}\log(1-\rho_{xu}^2),\qquad r_y=-\frac{1}{2}\log(1-\rho_{yv}^2),$$

define

$$G(r_x,r_y)=-\frac{1}{2}\log(1-\rho_{xy}^2\rho_{yv}^2\rho_{xu}^2).$$

Also define $G^*(r_x,r_y)$ using the quantities

$$\gamma=\rho_{xy}\rho_{xu}\rho_{yv},\qquad \beta=\rho_{xu}^2+\rho_{yv}^2-(1-\rho_{xy}^2)\rho_{xu}^2\rho_{yv}^2,$$ $$\rho=\frac{\beta}{2\gamma}-\sqrt{\left(\frac{\beta}{2\gamma}\right)^2-1},$$

and

$$G^*(r_x,r_y)=r_x+r_y+\frac{1}{2}\log\left[1+\frac{2\rho^2\gamma-\beta}{1-\rho^2}\right].$$

Gaussian-case conjecture. In the Gaussian case, the surfaces of $\mathcal{R}_{in}$ and $\mathcal{R}_{out}$ are

$$r_{in}(r_x,r_y)=G(r_x,r_y),\qquad r_{out}(r_x,r_y)=G^*(r_x,r_y).$$

These formulas are intended to provide explicit Gaussian inner and outer surfaces in parallel with the binary case. The supplied text does not state whether this claim has been proved or remains open.