The metric cotype gap conjecture

Let $(X,d)$ be a metric space, and let $q_X$ denote its metric cotype exponent. Metric cotype gap conjecture. There is no metric space $(X,d)$ for which

$$1<q_X<2.$$

The preceding results establish bounds for metric cotype on snowflake spaces and motivate this conjecture as a converse direction to the theorem for spaces bi-Lipschitz equivalent to ultrametric spaces. Whether metric cotype can take a value strictly between $1$ and $2$ remains open.