Residual-error conjecture for the recursive tridiagonal inversion algorithm

Let $A\in\mathbb{R}^{n\times n}$ be a nonsingular matrix whose inverse has only nonzero elements in the lower triangle. Let $X$ be the inverse computed numerically by algorithm (KWbasic), let $\varepsilon$ denote the machine precision, let $\operatorname{cond}(A)$ denote the condition number of $A$, and assume that $\varepsilon n\operatorname{cond}(A)<C$ for some constant $C$. Residual-error conjecture. The computed inverse satisfies

\max\left\\{\left\\|XA-I\right\\|_1,\left\\|AX-I\right\\|_1\right\\}\leq\varepsilon K(n)\operatorname{cond}(A),

for some $K(n)=O(n)$. This is obtained conditionally from the authors' preceding presumption about the computed inverse norm; the text says that a formal proof of that assumption may be difficult, so the claim remains open.