This is a follow-up of the previous post Law of the iterated logarithm in Hilbert space

The standard law of the iterated logarithm expresses that if $X_1, X_2, \ldots$ are iid real random variables with mean zero and variance $\sigma^2$, $$ \limsup_{n \to \infty} \frac {X_1 + \cdots + X_n}{\sqrt {2n \ln \ln n}} = \sigma $$ almost surely. Together with the same result for $-X_1, -X_2, \ldots$, the same limit holds true with absolute values around the sum $X_1 + \cdots + X_n$.

As indicated in the previous post by Iosif Pinelis, it is known (see the links in Law of the iterated logarithm in Hilbert space) that if $X, X_1, X_2, \ldots$ are iid random vectors in a separable Hilbert space $(H, \langle \cdot, \cdot \rangle, |\cdot |)$ with $E(X) = 0$ and $E(|X|^2) < \infty$, then $$ \limsup_{n \to \infty} \frac {|X_1 + \cdots + X_n|}{\sqrt {2n \ln \ln n}} = \sigma $$ almost surely where $$ \sigma = \sup \Big \{ \sqrt {E \big (\langle X, f \rangle ^2 \big) } : f \in H, |f| = 1 \Big\}. $$

Now the additional question is about the minimal assumption under which such a property holds. The real law of the iterated logarithm is known to be equivalent to $\sigma^2 = E(X^2) < \infty$. Does the law of the iterated logarithm in Hilbert space imply back that $E(|X|^2) < \infty$, or only $\sigma < \infty$ which seems weaker? Are there necessary and sufficient moment conditions for this law of the iterated logarithm in Hilbert space?