The Tarski alternative for Boolean inverse monoids

Let $S$ be a Boolean inverse monoid. An invariant mean on $S$ is a function $\mu\colon E(S)\to[0,1]$ such that $\mu(1)=1$, $\mu(e\vee f)=\mu(e)+\mu(f)$ when $ef=0$, and $\mu(s^{-1}s)=\mu(ss^{-1})$ for all $s\in S$.

The Tarski alternative. Exactly one of the following is true:

1. $S$ contains a copy of the Cuntz monoid $C_{2}$ as an inverse submonoid.
2. $S$ has an invariant mean.

This conjecture is motivated by work cited in the source and is an inverse-monoid analogue of a Tarski-type alternative. The two alternatives are mutually exclusive because a Boolean inverse monoid containing $C_{2}$ cannot possess an invariant mean; the source gives no resolution of whether they exhaust all possibilities.