Embeddings of Rearrangement Invariant Spaces that are not Strictly Singular

We give partial answers to the following conjecture: the natural embedding of a rearrangement invariant space E into L1([0,1]) is strictly singular if and only if G does not embed into E continuously, where G is the closure of the simple functions in the Orlicz space LΦ with Φ(x) = exp(x2)-1.