The binary-alphabet concave-envelope conjecture for Marton's inner bound

Let $(U,V)\to X\to(Y,Z)$ be a Markov chain for a broadcast channel with binary input alphabet $|\mathcal{X}|=2$. Define

$$T_\alpha(X):=\max_{p(u,v\mid x)}\alpha I(U;Y)+I(V;Z)-I(U;V),\qquad \alpha\geq1,$$

and let $\mathfrak{C}[f(X)]$ denote the upper concave envelope of $f$ at $p(x)$. Put $\bar\lambda=1-\lambda$.

Binary-alphabet concave-envelope conjecture. For all $\alpha\geq1$ and all such Markov chains,

$$-(\alpha-\bar\lambda)H(Y)-\bar\lambda H(Z)+T_\alpha(X) \leq \mathfrak{C}[-(\alpha-\bar\lambda)H(Y)-\bar\lambda H(Z)+\max\{\alpha I(X;Y),I(X;Z)\}].$$

The underlying pointwise inequality is known to fail for some binary-input broadcast channels, but the conjectured concave-envelope inequality may still hold for the distributions needed to compute the envelope. The source proves the underlying inequality for several cases and for the binary skew-symmetric broadcast channel, but gives no resolution of this conjecture.