The diagonal conjecture for primary pseudo-polynomials

Let $(a_n)_{n\geq 0}\in\mathbb{Z}^{\mathbb{N}}$ be a primary pseudo-polynomial, meaning that for every integer $n\geq 0$ and every prime number $p$,

$$a_{n+p}\equiv a_n \pmod p.$$

Its generating series is

$$f_a(x)=\sum_{n=0}^{\infty}a_nx^n.$$

A series is the diagonal of a rational fraction if it has the form

$$\sum_{n=0}^{\infty}u_{n,n,\ldots,n}z^n,$$

where the coefficients $u_{n_1,\ldots,n_k}$ come from a multivariate rational power series.

Diagonal conjecture. If the generating series of a primary pseudo-polynomial is the diagonal of a rational fraction, then $(a_n)_{n\geq 0}$ is a polynomial.

Algebraic series are diagonals of rational functions in two variables, so this extends the source's discussed algebraic-generating-series result to a broader class. The source presents the assertion as an interesting conjecture and gives no resolution.