Banach spaces $X$ with $L_2(X)$ isomorphic to $\ell_2(X)$

Question

I've recently come across this interesting thread

Banach spaces $X$ with $\ell_2(X)$ not isomorphic to $L_2([0,1],X)$

I'm interested in the opposite question. Are conditions known that ensure a Banach space $X$ to satisfy $L_2(X)\simeq \ell_2(X)$? For instance, do superreflexive spaces have this property?