The power-index conjecture for grid discrepancies

Let $d_n$ denote the discrepancy associated with the grid of size $n$. Power-index conjecture. Let $a$ be an odd natural number. If $a$ is not divisible by $21$, then for all $k\in\mathbb{N}$,

$$d_{a^k-1}=d_{a-1}.$$

The authors state that they are unaware of a proof; Goshima and Yamagishi conjectured a similar statement for tori when $a$ is prime.