Stability of symmetrically preconditioned conjugate gradient

Let $\boldsymbol{A}\in\mathbb{R}^{m\times n}$ be the least-squares matrix, and let $\boldsymbol{R}$ be computed from $\boldsymbol{A}$ by sketching and $\textsf{QR}$ factorizing the associated sketched system. Assume the hypotheses of the sketching-stability lemma. Apply the conjugate gradient algorithm to the resulting inner system. Stability of symmetrically preconditioned conjugate gradient. The conjugate gradient algorithm satisfies the stated inner-solver guarantee. This would provide a proof of backward stability for a version of sketch-and-precondition with iterative refinement using conjugate gradient rather than LSQR; the parser supplies no evidence that the claim has been proved or refuted.