Complementary Cusick conjecture on strict Hamming-weight increase

Let $t$ be a nonnegative integer, and define

$$\widetilde c_t=\operatorname{dens}\bigl\{n\in\mathbb N:w(n+t)>w(n)\bigr\},$$

where $\operatorname{dens} A$ denotes the asymptotic density of a set $A\subseteq\mathbb N$.

Complementary Cusick conjecture. For every nonnegative integer $t$,

$$\widetilde c_t\leq\frac12.$$

This is the statement complementary to Cusick's conjecture for strict rather than weak Hamming-weight increase. The source discusses it as a conjecture related to the Tu–Deng problem, and no resolution is supplied.