How many miles to βω?—Approximating βω by metric-dependent compactifications

It is known that the Stone–Čech compactification βX of a noncompact metrizable space X is approximated by the collection of Smirnov compactifications of X for all compatible metrics on X. We investigate the smallest cardinality of a set D of compatible metrics on the countable discrete space ω such that, βω is approximated by Smirnov compactifications for all metrics in D, but any finite subset of D does not suffice. We also study the corresponding cardinality for Higson compactifications.