Kitaoka's finiteness conjecture for universal ternary quadratic forms

A universal ternary classical quadratic form over a totally real number field is a positive definite classical quadratic form in three variables that represents every totally positive algebraic integer of the field. Kitaoka's finiteness conjecture. There are only finitely many totally real number fields admitting a universal ternary classical quadratic form. The paper proves this conjecture for totally real number fields of degree $4$; the general statement is presented as an influential conjecture and remains open.