The characterization of non-strictly singular embeddings into $L_1$

Let $E$ be a rearrangement invariant space on $[0,1]$. The space $G$ is the closure of the simple functions in the Orlicz space associated with the Orlicz function $e^{x^2}-1$. The natural embedding of $E$ into $L_1([0,1])$ is strictly singular if it is not an isomorphism on any infinite-dimensional subspace.

Embedding conjecture. The natural embedding $E\subset L_1([0,1])$ is not strictly singular if and only if $G$ embeds continuously into $E$.

The forward implication is the unresolved direction; the converse follows from the Rademacher-function characterization of continuous embeddings of $G$ into rearrangement invariant spaces, since the natural embedding is then an isomorphism on the span of the Rademacher functions. The conjecture seeks a complete criterion for when the embedding into $L_1$ fails to be strictly singular.