The uniform Galois-group conjecture for reductions of D-finite series

Let $f(x)\in\mathbb{Q}[[x]]$ be a $D$-finite series and let $n$ be the minimal order of a differential operator that annihilates $f(x)$. Let $K$ be a number field and let $(G_1,\ldots,G_t)$ be a family of subgroups of $\operatorname{GL}_n(K)$. For a prime $\mathfrak p$ of $K$, write $\mathcal{O}_{(\mathfrak p)}$ for the localization of the ring of integers of $K$ at $\mathfrak p$ and $k_\mathfrak p$ for its residue field. For an algebraic reduction $f(x)\bmod p$, its residual splitting field is the intersection of its splitting field in $\mathbb{F}_p(x)^{\mathrm{sep}}$ with $\overline{\mathbb{F}}_p$. Uniform Galois-group conjecture. There exist a number field $K$ and a family $(G_1,\ldots,G_t)$ of subgroups of $\operatorname{GL}_n(K)$ such that, for almost all rational primes $p$ for which $f(x)\bmod p$ is well defined and algebraic over $\mathbb{F}_p(x)$, and for any prime $\mathfrak p$ of $K$ above $p$, there exists $i\in\{1,\ldots,t\}$ such that the residual splitting field of $f(x)\bmod p$ is contained in $k_\mathfrak p$ and

$$\operatorname{Gal}\bigl(f(x)\mid k_\mathfrak p(x)\bigr)\simeq \operatorname{image}\bigl(G_i\cap\operatorname{GL}_n(\mathcal{O}_{(\mathfrak p)})\to\operatorname{GL}_n(k_\mathfrak p)\bigr).$$

The conjecture predicts that only finitely many linear groups govern the Galois groups of algebraic reductions, after allowing a fixed number field and residue-field reduction. The paper presents examples supporting this uniformity, but the general assertion remains open.