The countability conjecture for domable planar unit rhombi

Let $\mathcal A$ be the set of all $a\geq 0$ such that the planar rhombus $\rho(a,\sqrt{4-a^2})$ can be domed. Countability conjecture for $\mathcal A$. The set $\mathcal A$ is countable. This is supported in the paper by an implication from the closed-dome rigidity conjecture; the known inclusion of an infinite set only establishes that $\mathcal A$ is infinite.