The non-domability conjecture for the planar rhombus $\rho(1/2,\sqrt{15}/2)$

Let $\mathcal A$ be the set of $a\geq 0$ for which the planar rhombus $\rho(a,\sqrt{4-a^2})$ can be domed, and write $\rho_\lozenge:=\rho(1/2,\sqrt{15}/2)$. The $\rho_\lozenge$ conjecture. One has $1/2\notin\mathcal A$; equivalently, the planar unit rhombus $\rho_\lozenge$ cannot be domed. The conjecture is presented as an analogue of the $(2,2,1)$ triangle conjecture, and the paper notes that non-domability of the integral rhombus $2\rho_\lozenge$ would imply both claims.