Finite redundancy implies atomicity for reproducing pairs

Let $(\Psi,\Phi)$ be a reproducing pair on a measure space $(X,\mu)$, let $T_\Phi$ be the associated synthesis-type map, and define the redundancy by

$$R(\Phi):=\dim({\sf Ker}\\,T_\Phi).$$

Atomicity conjecture. If $R(\Phi)<\infty$, then $(X,\mu)$ is atomic.

This is proposed as an analogue of finite-redundancy results for frames, extending the connection between finite-dimensional kernels and atomic measure spaces to reproducing pairs. The paper does not provide evidence of a resolution.