Divisibility by 2 of Stirling numbers of the second kind and their differences

Let n, k, a and c be positive integers and b   be a nonnegative integer. Let ν2(k)ν2(k) and s2(k)s2(k) be the 2-adic valuation of k and the sum of binary digits of k  , respectively. Let S(n,k)S(n,k) be the Stirling number of the second kind. It is shown that ν2(S(c2n,b2n+1+a))⩾s2(a)−1ν2(S(c2n,b2n+1+a))⩾s2(a)−1, where 04k>4 is a power of 2, and δ(k)=0δ(k)=0 otherwise. This confirms a conjecture of Lengyel raised in 2009 except when k is a power of 2 minus 1.