Vanishing of second derived limits for nerves of open covers

Let $X$ be a locally compact separable metrizable space, and let $N_\alpha$ be the nerves of its open covers. Conjecture A. For each $n$,

$$\lim^2 H_n(N_\alpha)=0.$$

This conjecture asserts the vanishing of the second derived inverse limit of the homology groups of the nerves. Its status is not established in the supplied text.