Expanding eigenbasis conjecture for recursive eigenvector matrices

Let $A$ be a matrix of dimension at most $7$, and repeatedly construct eigenvector matrices $X_i$ whose columns have unit $\mathbb{L}_2$ norm. A matrix property holds for almost all matrices if, for every matrix $A$ and every $\epsilon>0$, there is a matrix $A'$ with $|A(i,j)-A'(i,j)|<\epsilon$ having that property. Expanding eigenbasis conjecture. For almost all matrices of dimension at most $7$, the recursively constructed matrices satisfy

$$X_i^\dagger X_i\to I$$

as $i$ tends to infinity. This conjecture concerns the limiting orthogonality of recursively computed eigenvectors and is supported in the paper by numerical observations and proofs for special cases; no resolution is supplied.