Algebraicity or infinite local monodromy for integral solutions at singular points

Let $X/\mathbb{C}$ be a smooth projective curve, let $D\subset X$ be a reduced effective divisor, and let $(\mathscr{E},\nabla:\mathscr{E}\to\mathscr{E}\otimes\Omega_X^1(\log D))$ be a logarithmic flat bundle on $(X,D)$. For $x\in D$, let $s$ be an integral formal flat section at $x$. Singular-point conjecture. Either $s$ is algebraic, or $(\mathscr{E},\nabla)$ has infinite monodromy at $x$. This is presented as a local prediction of the main conjecture; the source gives no resolution.