2-local Lazard correspondence

Definition
The 2-local Lazard correspondence is a slight variation of the usual Lazard correspondence that aims to establish a correspondence between things that 2-locally look like Lazard Lie rings and things that 2-locally look like Lazard Lie groups. Explicitly, it is a correspondence:

Lazard alternating rings $$\leftrightarrow$$ Lazard diassociative loops

The usual Lazard correspondence

Lazard Lie rings $$\leftrightarrow$$ Lazard Lie groups

is a subcorrespondence of this correspondence.

Observations
The first thing we observe is that under this correspondence, it is possible for Lie rings that are not Lazard Lie rings (but have small 2-local class) to get mapped to diassociative loops that aren't groups. Conversely, it is possible for groups that are not Lazard Lie groups (but have small 2-local class) to get mapped to alternating rings that are not Lie rings.