Christol's conjecture on globally bounded D-finite functions

A D-finite function is a formal power series satisfying a linear differential equation with polynomial coefficients, and a globally bounded function is one for which $S(Nx)\in\mathbb{Z}[[x]]$ for some positive integer $N$. A formal power series is a diagonal of a rational function if it is obtained by taking the diagonal coefficient sequence of a multivariate rational function.

Christol's conjecture. Every D-finite globally bounded function is the diagonal of a rational function.

This conjecture is cited as Christol's Conjecture 4 and is invoked here as a consequence of the implication $2\implies5$ for the class of P-recursive sequences considered in the paper; the supplied text does not state whether it has been resolved in full.