Chung–Goldwasser conjecture on maximal measurable 3-sum-free subsets of the unit interval

Let $A$ be a measurable 3-sum-free subset of $[0,1]$, meaning that $A$ contains no solution to $x+y=3z$. Let $4mu(A)4$ denote its Lebesgue measure. The sets $\mathcal A_1,\dots,\mathcal A_7$ are the seven sets obtained from

$$\mathcal A_0=\left(\frac{8}{177},\frac{4}{59}\right)\cup\left(\frac{28}{177},\frac{14}{59}\right)\cup\left(\frac{2}{3},1\right)$$

by adjoining one endpoint of each of the three intervals, except the endpoints $8/177$, $14/59$, and $2/3$.

Chung–Goldwasser conjecture. Every measurable 3-sum-free subset $A$ of $[0,1]$ satisfies

$$\mu(A)\leq\frac{77}{177}.$$

Moreover, if $\mu(A)=77/177$ and $A$ is maximal with respect to inclusion among the 3-sum-free subsets of $[0,1]$, then $A\in\{\mathcal A_1,\dots,\mathcal A_7\}$.

The conjecture identifies both the optimal Lebesgue measure and all inclusion-maximal extremal sets for the continuous 3-sum-free problem. The cases $k=1$ and $k\geq4$ were known, while the case $k=3$ was the remaining open case in the cited discussion.