Superfluity of the linear growth assumption for multiple correlation sequences

Let $(r_n)$ be the sequence of integers occurring in Theorems 4 and $4'$; those theorems establish equivalences for such sequences under a linear growth assumption $r_n=O(n)$.

Growth-assumption conjecture. In Theorems 4 and $4'$, the growth assumption on $(r_n)$ is superfluous.

The conjecture would extend the stated equivalences beyond sequences of at most linear growth. The authors explain that their proof relies on a theorem that is not helpful for faster-growing sequences, so the claim remains open in the source.