A Gaussian auxiliary-variable conjecture for the strengthened entropy power inequality

Fix $\alpha>0$. Let $X_1,X_2,W,W_2$ be mutually independent, with $W\sim N(0,1)$, $W_2\sim N(0,1-\alpha^2)$, and $X_1,X_2$ arbitrary random variables satisfying

$$\operatorname{Var}(X_1)=P_1,\qquad \operatorname{Var}(X_2)=P_2.$$

Define

$$Y_1=X_1+W,\qquad Y_0=\alpha Y_1+W_2,\qquad Y_2=Y_0+X_2.$$

Auxiliary-variable conjecture. There exists a random variable $V$ such that $Y_1\to Y_0\to V$ and

$$2^{2I(Y_1;V)}\geq\frac{2^{2I(X_1,X_2;Y_2)}}{P_2+1-\alpha^2},$$ $$2^{-2I(Y_0;V\mid Y_1)}\geq\frac{P_2\,2^{2I(X_1,X_2;Y_2)}}{(P_2+1-\alpha^2)(1+\alpha^2P_1+P_2)}.$$

This auxiliary-variable assertion is intended to provide the single-letter inequality needed in the strengthening of the entropy power inequality and the associated Gaussian interference-channel bounds. The supplied text does not state whether the conjecture has been proved or disproved.