Bézivin's conjecture on integral power-series solutions

Let $L\in\mathbb{Q}[x][\partial]$ be a differential operator. A basis of solutions of $Ly=0$ in $\mathbb{Z}[[x]]$ is a $\mathbb{Q}$-linearly independent basis when its elements are linearly independent over $\mathbb{Q}$.

Bézivin's conjecture. If $Ly=0$ has a $\mathbb{Q}$-linearly independent basis of solutions in $\mathbb{Z}[[x]]$, then these solutions are algebraic over $\mathbb{Q}(x)$.

This is presented as an apparently weaker statement than the Grothendieck $p$-curvature conjecture. The source does not report a resolution, so the conjecture remains open.