Conjecture on trivial differential Galois groups for the inhomogeneous equation

Let $P$ be a Picard–Vessiot extension of the field of rational functions over $\mathbb{C}$ obtained by adjoining solutions of the three displayed differential equations in the source, and let the parameters satisfy

$$\alpha,\beta\in\mathbb{Z}+\tfrac{1}{2},\qquad \alpha+\beta+r\in2\mathbb{Z},\qquad |\alpha-\beta|<r.$$

The trivial-Galois-group conjecture. However large $\alpha$, $\beta$, and $r$ are, the Galois group of the associated third-order homogeneous equation is trivial in the category of algebraic groups. The article proves membership in $P$ for certain parameter choices and reports numerical evidence for relatively small parameters, while the all-parameter assertion remains conjectural.