An abstract $L_1$ space is a Banach lattice $E$ such that $\|x+y\|=\|x\|+\|y\|$ for disjoint $x,y\in E$. The space $L_1[0,1]$ is a separable example that contains subspaces isomorphic to $L_p[0,1]$ for $1<p\leq 2$. The space of Borel measures $M[0,1]$ is a non-separable example, but each reflexive subspace of $M[0,1]$ is separable because $M[0,1]$ is isomorphic to the dual space of the separable space $C[0,1]$.

Given an uncountable set $I$, we endow $[0,1]^I$ with the product measure associated to the Lebesgue measure on $[0,1]$, and $L_1([0,1]^I)$ is a non-separable abstract $L_1$ space. Does it contain subspaces isomorphic to $L_p([0,1]^I)$ for $1<p\leq 2$?