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 \(\mat{S} \in \C^{n \times \ell}\), where \(k < \ell \ll n,\)  is there a matrix \(\mat{E}_{\delta,\mat{S}} \in \C^{n \times n} \) such that

  1. \(\hat{\mat{A}} := \mat{A} + \mat{E}_{\delta,\mat{S}}\) is SPSD,
  2. The top \(k\)-dimensional invariant subspace of \(\hat{\mat{A}}\) is also \(\mu\)-coherent,
  3. \(\|\hat{\mat{A}} – \mat{A}\|_2 < \delta \), and
  4. \(\mat{S}^t \hat{\mat{A}} \mat{S} \succeq \delta \cdot \mat{I}\) ?

It’d be fine if instead of (2) holding, the coherence of the top \(k\)-dimensional invariant subspace of \(\hat{\mat{A}}\) is slightly larger than that of \(\mat{A}\).