Stability conjecture for non-orthogonal coarse-grid correction in RN AMG

Let $A$ be nonsingular. Define the non-orthogonal coarse-grid correction using transfer operators

$$R={(P_{\textnormal{ideal}}^{AA^*})}^T,\qquad P=P_{\textnormal{ideal}}^{A^*A}.$$

The correction is stable in the sense that

$$\left\|P(RAP)^{-1}RA\right\|_{\sqrt{A^*A}}=\left\|I-P(RAP)^{-1}RA\right\|_{\sqrt{A^*A}}=C,$$

for some constant $C$ independent of mesh spacing. This stability would, together with the preceding approximation result, support two-grid convergence of the root-node based algebraic multigrid method for nonsymmetric problems; the supplied text does not establish the conjecture or indicate that it has been resolved.