Critical-orbit convergence conjecture for non-linearizable indifferent fixed points

Let $f$ be a rational function of degree $2$ or more, with an indifferent irrational non-linearizable fixed point $z_0$. A point $c_0$ is a critical point if it is a critical point of $f$.

Critical-orbit convergence conjecture. There exists a critical point $c_0$ of $f$ such that

$$\lim_{n\to +\infty} \frac1n \sum_{j=0}^{n-1} \delta_{f^j(c_0)} \to \delta_{z_0}.$$

The conjecture is proposed as a replacement for Lyubich's earlier conjecture that a critical orbit converges directly to an indifferent irrational non-linearizable fixed point, which the source states is false for a generic rational function. It asks instead for convergence of the empirical measures of a critical orbit to the point mass at the fixed point; its status is not established in the supplied text.