Sum-rate conjecture for the binary skew-symmetric broadcast channel

Let $(U,V)$ be auxiliary random variables and let $X,Y_1,Y_2$ be the input and outputs of the binary skew-symmetric channel with $p=\frac{1}{2}$, such that $(U,V) \to X \to (Y_1,Y_2)$ is a Markov chain. Sum-rate conjecture. The following inequality holds:

$$I(U;Y_1) + I(V;Y_2) - I(U;V) \leq \max\{I(X;Y_1), I(X;Y_2)\}.$$

If true, this conjecture would bound the sum rate of the Márton inner bound by approximately $0.3616$, while the evaluated outer bound has maximum sum rate approximately $0.3711$. Thus it would establish that the inner and outer bounds are not tight for the binary skew-symmetric channel.