The apparent-singularity conjecture for the associated third-order ODE

Consider the third-order ODE

$$y”'-\left(\frac{\alpha}{z^2}+B\right)y'+\left(\frac{-2\beta}{z^3}+\frac{D_0}{z^2}+D\right)y=0.$$

The point $0$ is an apparent singularity when the associated local monodromy is trivial. Let $\hat P_{0,1},\hat P_{0,2},\hat P_{0,3}$ be the polynomial conditions obtained from the local Frobenius analysis, so that the corresponding polynomial system is

$$\hat P_{0,1}(B,D_0,D)=\hat P_{0,2}(B,D_0,D)=\hat P_{0,3}(B,D_0,D)=0.$$

Apparent-singularity conjecture. The point $0$ is an apparent singularity if and only if $B=D_0=D=0$; equivalently, this polynomial system has only the trivial solution $B=D_0=D=0$. This claim is presented as a further conjectural assertion about the reduced polynomial system; the supplied text gives no resolution.