Conjecture on singularities from three formal solutions of a third-order ODE

Let $a,b,c,d$ be analytic functions at $x_0 otin\mathbb R$, and consider the linear homogeneous ordinary differential equation

$$a(x)y^{\prime\prime\prime}+b(x)y^{\prime\prime}+c(x)y^\prime+d(x)y=0.$$

Suppose that this equation admits three linearly independent formal solutions $y_1(x),y_2(x),y_3(x)$. Three-solution singularity conjecture. Then $x_0$ is an ordinary point or a regular singular point for the ODE, and the solutions are analytic and convergent. The claim would characterize ordinary and regular singular points through the existence of a full set of formal solutions; its status is not established by the supplied text.