The Elementary Type Conjecture for finite quaternionic structures

Let $S$ be a finite quaternionic structure. It 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.

Elementary Type Conjecture. Every finite quaternionic structure is of elementary type.

The conjecture is the quaternionic-structure formulation of the prediction that all finitely generated Witt rings arise from the listed elementary building blocks. The source reports that the lack of counterexamples motivated it, but the supplied status evidence indicates that it has been disproved.