The minimal-degree conjecture for triquadratic fields

Let $L = \mathbb{Q}(\sqrt{A}, \sqrt{B}, \sqrt{C})$ for distinct square-free non-zero integers $A$, $B$, and $C$, and let $M = \mathbb{Q}(\sqrt{A}, \sqrt{B})$. Here $\min\deg_L(v)$ denotes the minimum degree over $L$ among primitive-element representations of $v$ as considered in the paper. Minimal-degree conjecture. There exists a rational number $a$ such that

$$\min\deg_L(\sqrt{A} + a\sqrt{B}) > [L:M].$$

This conjecture asserts the existence of an element in every triquadratic field whose minimal degree differs from the corresponding field-extension degree. The preceding examples and results establish instances of the phenomenon, but the general assertion remains open.