Free Idempotent Rigs and Monoids

I’ve been having a lot of fun on Mathstodon lately, and here’s an example.

A rig R has a commutative associative addition, an associative multiplication that distributes over addition, an element 0 with r+0 = r and for all r ∈ R and an element 1 with 1r = r = r1 for all r ∈ R

A rig is idempotent if r r = r for all r ∈ R.

Is the free idempotent rig on 2 generators finite? If so, how many elements does it have?

Morgan Rogers raised this issue on the Category Theory Community server, and after a ...