Normal-operator extension of the normalized-frame nonexistence theorem

Let $A$ be a normal operator on a Hilbert space $\mathcal{H}$, and let $\mathcal{G}$ be the set of vectors used to form the normalized system

$$\left\{\frac{A^n g}{\lVert A^n g\rVert}\right\}_{g\in\mathcal{G},\;n\geq 0}.$$

Normalized-frame conjecture. The system above is not a frame for $\mathcal{H}$.

The corresponding assertion is known for self-adjoint operators, while the conjecture is false for non-normal operators; hence this proposed extension to all normal operators has been refuted.