The weak Elementary Type Conjecture for field-realized quaternionic structures

Let $S$ be a finite quaternionic structure realized by a field, meaning that it is the quaternionic structure associated with a field. Such a structure is of elementary type if it can be obtained from ${\mathbb{L}}_0$, ${\mathbb{L}}_{1,0}$, ${\mathbb{L}}_{1,1}$, ${\mathbb{L}}_1$, and local type structures using only direct products and group extensions.

Weak Elementary Type Conjecture. Every finite quaternionic structure that is realized by a field is of elementary type.

This is presented as a possibly weaker version of the Elementary Type Conjecture because it is not known whether every quaternionic structure is realized by a field. Its resolution status is not specified in the source.