The non-domability conjecture for two unit triangles

Let $\Delta_1,\Delta_2\subset\mathbb R^3$ be unit triangles, and let $\Upsilon=\Delta_1\cup\Delta_2$. Two-triangle non-domability conjecture. There exist unit triangles $\Delta_1$ and $\Delta_2$ such that $\Upsilon$ cannot be domed. This is posed as a multi-curve analogue of the problem of finding curves that cannot be domed, with particular relevance to the Steinhaus problem.