Let s be the sum-of-digits function in base 2, which returns the number of 1s in the base-2 expansion of a nonnegative integer. For a nonnegative integer t, define c_t as the asymptotic density of nonnegative integers n such that s(n+t)≥s(n). T. W. Cusick conjectured that c_t>1/2.  We have the elementary bound 0<c_t<1; however, no bound of the form 0<α≤c_t or c_t≤β <1, valid for all t, is known. In this paper, we prove that c_t>1/2−ε as soon as t contains sufficiently many blocks of 1s in its binary expansion. In the proof, we provide estimates for the moments of an associated probability distribution; this extends the study initiated by Emme and Prikhod’ko (2017) and pursued by Emme and Hubert (2018).