Expanding orthogonal similarity variant conjecture

Let $X_1$ be an initial matrix, let $Z_1$ be its eigenvector matrix, and for each subsequent step let $Z_i$ be the eigenvector matrix of $Y_{i-1}$. Let $Q_i$ be a random orthogonal or unitary matrix, and define

$$Y_i=Q_iZ_iQ_i^\dagger.$$

Expanding orthogonal similarity variant conjecture. For almost all matrices of dimension at most $7$, repeatedly applying this eigenvector construction and random orthogonal or unitary similarity transformation gives

$$Y_i^\dagger Y_i\to I$$

as $i$ tends to infinity. This is a variant of the expanding eigenbasis claim involving random similarity transformations; the paper presents it as a conjecture supported by numerical evidence, with no resolution supplied.