The cotype-2 threshold-embedding conjecture

Let $X$ be a Banach space of cotype $2$, and let $S\subseteq X$. The subset $S$ threshold-embeds into Hilbert space if there is a threshold embedding of $S$ into a Hilbert space. Cotype-2 threshold-embedding conjecture. $S$ threshold-embeds into Hilbert space if and only if $S$ has Markov type $2$. This is proposed as the first step toward a nonlinear analogue of Kwapien's theorem; the corresponding assertion is established in the paper for subsets of Banach spaces that admit a uniform embedding into a Hilbert space, while the cotype-$2$ case remains open.