Cusick's conjecture on the sum-of-digits correlation measure

Let $t\geq 0$ be an integer, and let $\mu_t$ be the probability measure defined by the limiting distribution of $\mathsf s(n+t)-\mathsf s(n)$. Set

$$c_t\coloneqq\sum_{j\geq 0}\mu_t(j).$$

Cusick's conjecture. For all $t\geq 0$ we have

$$c_t>\frac12.$$

This conjecture concerns the positive-side mass of the distribution of binary sum-of-digits differences and remains open in the source.