Are there perturbations that preserve incoherence and give nicely conditioned “submatrices”?

Let $$\mat{P}_{\mathcal{U}}$$ denote the projection unto a $$k$$-dimensional subspace of $$\C^{n}.$$ We say $$\mathcal{U}$$ is $$\mu$$-coherent if $$(\mat{P}_{\mathcal{U}})_{ii} \leq \mu \frac{k}{n}$$ for all $$i = 1, \ldots, n.$$ Let $$\mat{A} \in \C^{n \times n}$$ be a SPSD matrix whose top $$k$$-dimensional invariant subspace is $$\mu$$-coherent. Given $$\delta > 0$$ and \(\mat{S} \in \C^{n \times [...]