The characterization of admissible Banach spaces by nontrivial type

A Banach space $E$ is called admissible when it satisfies the admissibility condition defined in the paper. A Banach space has nontrivial type if its type is greater than $1$.

Admissibility conjecture. A Banach space is admissible if and only if it has nontrivial type.

The preceding lemma shows that Banach spaces with trivial type are not admissible, and the conjecture asserts that this is the only obstruction. The supplied text gives no resolution of the converse implication, so the characterization remains open.