Inverse bound conjecture for arbitrary-norm rank-revealing QR factorization

Let $A\in\mathbb{R}^{m \times m}$, let $\\|\cdot\\|$ be a norm on $\mathbb{R}^m$, and let $P,Q,R$ be the factors output by the arbitrary-norm rank-revealing QR algorithm described in the paper. Inverse bound conjecture. There exists a constant $C_2>0$, depending only on the norm $\\|\cdot\\|$, such that

$$\min_{\\|x\\|=1}\\|Qx\\|\geq C_2.$$

Together with the theorem's upper bound on $Q$, this would show that the factor $Q$ is uniformly well-conditioned with respect to the chosen norm, independently of $A$. The paper reports numerical evidence for the claim, but states that a complete proof remains elusive.