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

Let $fin \mathbb{Q}[[z]]$ be a globally bounded and D-finite power series, and let $L^{\mathrm{min}}_f$ denote its minimal annihilating differential operator. Christol–André conjecture. The following assertions should hold: (1) $f$ is the diagonal of a rational function; (2) if $z=0$ is an ordinary point for $L^{\mathrm{min}}_f$, then $f$ is algebraic; and (3) if the monodromy of $L^{\mathrm{min}}_f$ at $z=0$ is semisimple, equivalently if $z=0$ is not a logarithmic singularity of $L^{\mathrm{min}}_f$, then $f$ is algebraic. These conjectures would give structural and effective algebraicity criteria for globally bounded D-finite series. The first assertion is the diagonal conjecture, while the latter two provide algebraicity under additional local hypotheses.