The generalized factorization conjecture for Marton's inner bound

For a broadcast channel, define

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

For a product channel define $T_\alpha(X_1,X_2)$ analogously, and let $\mathfrak{C}[f(X)]$ be the upper concave envelope of $f$ at $p(x)$. Put $\bar\lambda=1-\lambda$.

Generalized factorization conjecture. For all product channels, all $\lambda\in[0,1]$, all $\alpha\geq1$, and all $p(x_1,x_2)$,

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

This equivalent formulation is sufficient for proving optimality of the entire private-message region in Marton's inner bound. The source gives no resolution of the generalized conjecture.