Conjecture on algebraicity, diagonals and global boundedness for linear-coefficient recurrences

Let $(s_n)_n$ be a sequence of rational numbers satisfying a linear homogeneous recurrence relation with polynomial coefficients of degree $1$:

$$\sum\limits_{k=0}^{d}(a_kn+b_k)s_{n-k}=0,\qquad a_i\ne0\text{ for }i=0,\ldots,d,\qquad d\in\mathbb{N}.$$

Let $S(x)=\sum\limits_{n\geq0}s_nx^n$ be its generating function.

The algebraicity–diagonal–global-boundedness conjecture. The following statements are equivalent:

1. $S(x)$ is algebraic.
2. $S(x)$ is a diagonal.
3. $S(x)$ is globally bounded.

This is proposed as a possible generalization of the paper's theorem. The supplied text gives no resolution or partial results for this conjecture.