Monotonicity of $(k,\beta)$-Poissonian box correlations

Let $0<\beta_1<\beta_2\leq 1$ and let $k\geq 2$. A sequence $x=(x_n)_{n\in\mathbb{N}}$ of real numbers has $(k,\beta)$-Poissonian box correlations if it satisfies the corresponding $(k,\beta)$-Poissonian box-correlation property.

Monotonicity conjecture. If $x$ has $(k,\beta_2)$-Poissonian box correlations, then it also has $(k,\beta_1)$-Poissonian box correlations.

This would extend the corresponding implication for Poissonian correlations and is expected to follow from the proposition relating Poissonian box correlations to the functions defining them. The proof of the analogous result does not transfer to this more general setting, and no alternative rigorous argument is known.