Non-subnormality conjecture for Cauchy duals on three-point atomic spaces

Let $\zeta_1,\zeta_2,\zeta_3$ be any three distinct points on the unit circle, let $c_1,c_2,c_3$ be positive real numbers, and set

$$\mu=c_1\delta_{\zeta_1}+c_2\delta_{\zeta_2}+c_3\delta_{\zeta_3}.$$

Let $M_z$ denote multiplication by $z$ on $D(\mu)$, and let $M_z'$ be its Cauchy dual. Non-subnormality conjecture. The Cauchy dual $M_z'$ of $M_z$ on $D(\mu)$ is not subnormal.

The conjecture extends the paper's example from a particular three-point measure to arbitrary positive weights supported at three distinct points of the unit circle. The authors report strong evidence for the claim, while a proof in this generality remains open.