Extension of the law of iterated logarithm from basis vectors to all vectors

Let $T$ be the random weighted shift associated with independent weights $\{X_n\}_{n=1}^{\infty}$ on a Hilbert space $\mathcal{H}$, and let $\{e_k\}_{k=1}^{\infty}$ be its associated orthonormal basis. Assume $\mathbb{E}(\ln X_1)=0$ and $\mathbb{E}((\ln X_1^2)^2)=\sigma^2<\infty$. Lemma gives, for each basis vector $e_k$, the almost-sure law-of-iterated-logarithm limits for $\ln\|T^ne_k\|^2$. Extension conjecture. Lemma holds for every $x\in\mathcal{H}$, instead of only for $e_k\in\mathcal{H}$. This would extend the precise almost-sure growth-rate statement from individual basis vectors to arbitrary Hilbert-space vectors.