Finiteness conjecture for virtually solvable telescoper Galois groups

Let $T$ be a minimal-order telescoper for an integral

$$\int a(x,h)\,dx$$

where $a$ is algebraic in $x$ and $h$. Finiteness conjecture. If the Galois group of $T$ is virtually solvable, then it is finite.

This conjecture concerns the differential Galois groups arising from telescopers for algebraic integrands. It would distinguish the finite virtually solvable groups from the non-finite virtually solvable groups in this setting; the source presents it as an open question motivated by the difficulty of computing such Galois groups in general.