On the ABK conjecture, alpha-well quasi orders and Dress-Schiffels product

The following is a 2008 conjecture from [3] [ABK Conjecture] Every well quasi order (wqo) is a countable union of better quasi orders (bqo). We obtain some partial progress on the conjecture, in that we show that the class of orders that are a countable union of better quasi orders (sigma-bqo) is closed under various operations. These include diverse products, such as the little known but natural Dress-Shieffels product. We develop various properties of the latter. In relation with the main question, we explore the class of alpha-wqo for countable ordinals alpha and obtain several closure properties and a Hausdorff-style classification theorem. Our main contribution is the discovery of various properties of sigma-bqos and ruling out potential counterexamples to the ABK Conjecture.