On Kitaoka's conjecture and lifting problem for universal quadratic forms

For a totally positive definite quadratic form over the ring of integers real number field $K$, we show that there are only finitely many extensions $K$ fixed degree which is universal (namely, those have short basis in suitable sense). Along way give general construction rank bounded by $D(\log D)^{d-1}$, where $d$ $\mathbb Q$ and $D$ its discriminant. Furthermore, for any prove (weak) Kitaoka's conjecture fields with ternary form.