The GL2 characterization by finite-cotype lattice embeddings

Let $X$ be a $K$-convex Banach space. The property $\mathrm{GL}_2$ is the Gordon–Lewis property with exponent $2$, and a Banach lattice has finite cotype in the usual Banach-space sense. GL2 characterization conjecture. The space $X$ has $\mathrm{GL}_2$ if and only if both $X$ and its dual $X^*$ embed into Banach lattices of finite cotype. The source presents this as the second of two conjectures concerning embeddings into Banach lattices of finite cotype and supplies no resolution.