Cusick's simplified conjecture for paired binary shifts

Let $s$ be the binary sum-of-digits function, and let $c_t$ be the asymptotic density

$$c_t=\operatorname{dens}\{n\geq 0:s(n+t)\geq s(n)\}.$$

For $t\geq 0$, choose $\lambda$ such that $2^\lambda\leq t<2^{\lambda+1}$, and define

$$t'=3\cdot 2^\lambda-t.$$

Cusick's simplified conjecture. For all $t\geq 0$, one has $c_t+c_{t'}>1$; equivalently, at least one of $t$ or $t'$ satisfies Cusick's conjecture. This is presented as an easier substitute for the original conjecture, whose full statement remains open; the paper proves only asymptotic versions and partial results.