Continuous-frame interval invariance for normal reductive operators

Let $A\in\mathcal{B}(\mathcal{H})$ be an invertible self-adjoint operator and let $\mathcal{G}$ be a countable set in $\mathcal{H}$. Theorem asserts that

$$\{A^t g\}_{g\in\mathcal{G},t\in[0,1]}$$

is a semi-continuous frame in $\mathcal{H}$ if and only if

$$\{A^t g\}_{g\in\mathcal{G},t\in[0,L]}$$

is a semi-continuous frame in $\mathcal{H}$ for every finite positive $L$.

Normal-reductive extension conjecture. Theorem remains true if $A$ is a normal reductive operator.

The conjecture proposes extending the interval-invariance result from invertible self-adjoint operators to normal reductive operators. The supplied text gives no resolution, so its status remains open.