Universality conjecture for the 34 listed ternary lattices

Consider the $34$ ternary lattices listed in the theorem immediately preceding the conjecture, over the relevant real quadratic fields. A universal ternary lattice is a ternary quadratic lattice representing every totally positive element of the ring of integers. Universality conjecture. All the $34$ ternary lattices listed in the above theorem are indeed universal. The source gives computational verification of representability bounds for these lattices, but a proof of universality for all $34$ lattices remains open.