Inverse Power Method conjecture on high-accuracy evaluation of the first eigenvector coordinate

Let $A$ be the symmetric tridiagonal matrix considered in the paper, let $\lambda$ be an eigenvalue of $A$, and let

$$X=(X_1,\dots,X_n)\in\mathbb{R}^n$$

be its unit-length eigenvector. Let $\varepsilon>0$ denote the machine precision, and suppose that $\lambda$ is known with relative error approximately $\varepsilon$, that $\lambda-A_1>2$, and that $K>0$ is an integer satisfying

$$K>\frac{\log\left(|X_1|\right)}{\log(\varepsilon)}+1.$$

Inverse Power Method conjecture. After $K$ iterations of the Inverse Power Method, the coordinate $X_1$ is evaluated with high relative accuracy; more specifically, its relative accuracy is of the same order of magnitude as that obtained by the algorithms described earlier in the paper.

The claim concerns the numerical accuracy of an iterative method for computing eigenvectors of symmetric tridiagonal matrices. It is supported by extensive numerical experiments, but the supplied source does not state a proof or resolution, so its status remains open.