Cyclic cube-free density conjecture

Let $\mathbb{Z}_N$ be the cyclic group of order $N$, and let $A\subset\mathbb{Z}_N$ be $d$-cube-free, meaning that there is no multiset $S=\{a_1,\ldots,a_d\}$ such that

$$\Sigma^*S=\left\{\sum_{i\in I}a_i:\varnothing\neq I\subset [d]\right\}\subset A.$$

Cyclic cube-free density conjecture. If $d\mid N$, then

$$|A|\leqslant \frac{d-1}{d}N.$$

The bound is sharp: when $N$ is divisible by $d$, the set $\{b\in\mathbb{Z}_N:b\equiv 1,2,\ldots,d-1\pmod d\}$ has this size. The paper proves the case $d=3$, while the general assertion remains open in the supplied text.