The general-order linear ODE symmetry-generator conjecture for the group $G_c$

Consider the general linear ordinary differential equation

$$y^{(n)} + a^{n-1} y^{(n-1)} + a^{n-2} y^{(n-2)} + \dots + a^0 y=0,$$

with $n\geq 3$. Let $G_S$ be its symmetry group, with infinitesimal generator $X=\{\xi,\eta,\phi\}$, where the linearity of the equation gives $\eta=hy+g$, and let $G_c$ denote the group obtained by imposing $g=0$. Define

$$X^0=\{\xi,\phi\}_{\vert_{g=0}}.$$

General-order linear ODE symmetry-generator conjecture. The vector field $X^0$ is the infinitesimal generator of $G_c$.

The claim extends a result established in the source for orders $n=3,4,5$ to the general linear equation of arbitrary order. Its status is not resolved in the supplied context.