Cyclic progression-free density conjecture

Let $\mathbb{Z}_N$ be the cyclic group of order $N$, and let $A\subset\mathbb{Z}_N$ be $\{x,2x,\ldots,(d-1)x\}$-free, meaning that no $x$ has all of $x,2x,\ldots,(d-1)x$ in $A$. Cyclic progression-free density conjecture. If $d\mid N$, then

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

The conjecture is motivated by generalizing the proof of the $d=3$ cube case; the supplied text gives no resolution status.