Conjectured existence of universal ternary lattices over thirteen real quadratic fields

Let $D$ be a positive squarefree integer, let

$$F=\mathbb{Q}(\sqrt{D}),$$

and let $\mathcal{O}_F$ denote the ring of integers of $F$. A universal ternary $\mathcal{O}_F$-lattice is a universal quadratic lattice of rank $3$ over $\mathcal{O}_F$. Existence conjecture. If

$$D\in\{2,3,5,6,7,10,13,17,21,33,41,65,77\},$$

then there exists a universal ternary $\mathcal{O}_F$-lattice. The preceding theorem proves that these are the only possible values of $D$ for real quadratic fields, but the converse remains conjectural.