Existence of diagonal universal ternary forms

Determine whether there exist diagonal (i.e., diagonalizable) positive definite ternary quadratic forms over the ring of integers of any totally real number field that are universal, or prove that no such diagonal universal ternary form exists beyond known cases.

Background

Diagonal forms are a natural and historically significant subclass of quadratic forms. While the sum of three squares is known to be universal over Q(√5), general results of Siegel constrain universality of sums of squares; nevertheless, the broader existence of diagonal ternary universal forms remains unsettled.

The authors explicitly state that the question of (non)existence of diagonal universal ternary forms is still open, highlighting a fundamental gap in the understanding of universal forms under diagonal restrictions.