Haralambis' equality conjecture for three-element distance sets

Let $D$ be a three-element set of non-negative integers. Let $\kappa(D)$ be the supremum, over real $t$, of the smallest distance from $td$ to an integer as $d$ ranges over $D$, and let $\mu(D)$ be the supremum of the densities of sequences whose pairwise differences do not lie in $D$.

Haralambis' equality conjecture. Every three-element $D$-set satisfies

$$\kappa(D)=\mu(D).$$

The conjecture is presented in the conclusion as a particularly interesting problem because exact computation of $\mu(D)$ is difficult in general. The supplied text gives no resolution status.