Ruzsa's analogue for alpha-primary pseudo-polynomials

Let $0<\alpha<1$, let $\mathcal P$ be a set of primes satisfying

$$\prod_{\substack{p\in\mathcal P\p\le j}}p=e^{\alpha j}+o(e^{\alpha j}),$$

and let $f:\mathbb N\to\mathbb Z$ be an $\alpha$-primary pseudo-polynomial, meaning that

$$f(n+p)\equiv f(n)\pmod p$$

for every $p\in\mathcal P$ and $n\in\mathbb N$.

The alpha-primary growth conjecture. If $f(n)=O(c^n)$ for some $c<e^\alpha$, then $f$ is a polynomial.

The paper constructs a non-polynomial $\alpha$-primary pseudo-polynomial with growth $O(e^{(\alpha+\epsilon)n})$, so the conjecture identifies $e^\alpha$ as the expected critical exponential rate. No resolution is given here.