The non-domability conjecture for the isosceles triangle with sides $(2,2,1)$

Let $\Delta$ be an isosceles triangle with side lengths $(2,2,1)$. The $(2,2,1)$ triangle conjecture. The triangle $\Delta$ cannot be domed. This conjecture is a central obstruction in the paper: if it were false, every integer-sided triangle could be domed, and a flexible Bricard octahedron would result.