Christol–André conjecture on globally bounded D-finite power series

Let $y(x)=\sum_{k\geq 0}a_kx^k\in\mathbb{Q}[[x]]$ be D-finite. Assume that the sequence $(a_k)_{k\geq 0}$ has at most geometric growth, that there exists $N\in\mathbb{N}$ such that $y(Nx)-y(0)\in\mathbb{Z}[[x]]$, and that, in the minimal-order monic linear differential equation satisfied by $y(x)$, $x=0$ is not a pole of any coefficient $a_i(x)$. Christol–André conjecture. Then $y(x)$ is algebraic. This conjecture proposes an arithmetic strengthening of Eisenstein's theorem: suitable integrality and growth conditions, together with a regularity condition at the origin, should force a D-finite series to be algebraic. The source presents it as open; it also mentions a variant replacing the third condition by the absence of logarithms in the local solutions at $x=0$.