A sharper loss-of-orthogonality bound for MGS-HA

Let $A$ be the symmetric positive-definite matrix defining the non-standard inner product, let $Z$ be the input matrix with $m$ rows, let $n$ be the number of columns, and let ${\bf u}$ denote the unit roundoff. Write $\widehat Q$ for the computed $A$-orthogonal factor and let $\kappa(\cdot)$ denote the condition number. The loss of $A$-orthogonality is measured by $\|\widehat Q^{\rm T}A\widehat Q-I_n\|$. MGS-HA orthogonality-bound conjecture. The loss of $A$-orthogonality of MGS-HA can be bounded as

$$\|\widehat Q^{\rm T} A \widehat{Q} - I_n \| \leq \frac{\mathcal{O}(m^{3/2}) {\bf u} \left( \kappa(A) + \kappa(A^{1/2}Z) \right)}{1-\mathcal{O}(m^{3/2}) {\bf u} \left( \kappa(A) + \kappa(A^{1/2}Z)\right)} \approx \mathcal{O}(m^{3/2}) {\bf u} \left( \kappa(A) + \kappa(A^{1/2}Z) \right).$$

The quantity on the right is motivated by numerical experiments, where it describes the observed loss of $A$-orthogonality for MGS-HA more sharply than the earlier bound, although the paper states that it has no theoretical background yet.