Congruence classes of 2-adic valuations of Stirling numbers of the second kind

We analyze congruence classes of $S(n,k)$, the Stirling numbers of the second kind, modulo powers of 2. This analysis provides insight into a conjecture posed by Amdeberhan, Manna and Moll, which those authors established for $k\le 5$. We provide a framework that can be used to justify the conjecture by computational means, which we then complete for $k=5,6,\mathrm{.}\mathrm{.}\mathrm{.},20$.