Feichtinger's conjecture for kernel functions

Let $\mathcal{H}$ be a reproducing kernel Hilbert space, and for each point $z$ let $k_z^{\mathcal{H}}$ be its kernel function and $\widetilde{k}_z^{\mathcal{H}}$ the corresponding normalized kernel function. A sequence of vectors is Bessel if its analysis coefficients are square-summable with a uniform bound, and a Riesz basic sequence if it is a Riesz sequence forming a basis for its closed linear span. Feichtinger Conjecture for Kernel Functions. Every Bessel sequence of normalized kernel functions in every reproducing kernel Hilbert space can be partitioned into finitely many Riesz basic sequences. This is a reproducing-kernel formulation of the Feichtinger conjecture, and the supplied text presents it as an equivalent formulation without resolving it.