Limiting proportions of sum-dominant, difference-dominant, and balanced sets

Let $P$ be any arithmetic progression with length $n$. Define the proportions

$$\rho_- = \lim_{n\to\infty}2^{-n}\#\{S\subseteq P:S\text{ is difference-dominant}\},$$ $$\rho_+ = \lim_{n\to\infty}2^{-n}\#\{S\subseteq P:S\text{ is sum-dominant}\},$$

and

$$\rho_= = \lim_{n\to\infty}2^{-n}\#\{S\subseteq P:S\text{ is sum-difference-balanced}\}.$$

Limiting-proportion conjecture. The three limiting proportions all exist and are positive.

The preceding theorem establishes positive lower bounds for the corresponding proportions for every sufficiently large $n$, so only the existence of the limits is conjectural. The claim concerns the asymptotic distribution of the three types of subsets of an arithmetic progression.