The block-appending conjecture for the sum-of-digits correlation measure

Let $t\geq 1$ be an odd integer, and let $c_t$ denote the correlation quantity associated with the binary sum-of-digits difference for $t$. For each $K\geq 1$, consider the integer $2^Kt-1$ and its corresponding quantity $c_{2^Kt-1}$. Block-appending conjecture. For each odd integer $t\geq 1$ we have

$$c_t\geq\lim_{K\to\infty}c_{2^Kt-1}.$$

The conjecture formalizes replacing the rightmost $1$ in the binary expansion of an odd integer by a $0$ followed by infinitely many $1$s. It is relevant because results cited in the source show $c_t>1/2$ when $t$ contains sufficiently many blocks of $1$s, but the conjecture itself remains open in the source.