Tightness conjecture for the MAP conditional-entropy bound

Let $h_n$ denote the normalized conditional entropy in the code ensemble, and let $\tau$ and $\rho$ be the degree sequences appearing in Theorem 1. For each admissible random variable $V$, let $\phi_V$ be the corresponding trial-entropy functional. Under the hypotheses of Theorem 1, tightness conjecture.

$$\lim_{n\to\infty}h_n=\sup_V\\,\phi_V,$$

where the supremum is taken over the space of admissible random variables, with the degree sequences used as arguments of $\phi_V$ fixed to be those of Theorem 1. The conjecture asserts that the upper bound from Theorem 1 is asymptotically tight; establishing equality requires optimizing the trial entropy over all admissible random variables.