The factorization conjecture for Marton's inner bound

Let $\mathfrak{q}(y,z\mid x)$ be a broadcast channel. Define

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

and let $\mathfrak{C}[f(X)]$ denote the upper concave envelope of $f$ at $p(x)$. For a product of two broadcast channels, write $T(X_1,X_2)$ for the analogous quantity with outputs $(Y_1,Y_2)$ and $(Z_1,Z_2)$, and put $\bar\lambda=1-\lambda$.

Factorization conjecture. For all product channels, all $\lambda\in[0,1]$, and all $p(x_1,x_2)$,

$$-\lambda H(Y_1,Y_2)-\bar\lambda H(Z_1,Z_2)+T(X_1,X_2)$$ $$\leq \mathfrak{C}[-\lambda H(Y_1)-\bar\lambda H(Z_1)+T(X_1)] +\mathfrak{C}[-\lambda H(Y_2)-\bar\lambda H(Z_2)+T(X_2)].$$

The conjecture would imply that Marton's inner bound is optimal for the sum-rate of two-receiver discrete memoryless broadcast channels. It has been established for the endpoint values of $\lambda$, when one component channel is deterministic, and when one receiver in a component is more capable than the other; the general statement was subsequently established in the cited work.