Unbounded maximal elements in the Frobenius problem over real number fields

Let $K$ be a real number field and let $\beta_1,\dots,\beta_d\in\mathfrak{O}_K^+$ be a basis for $\mathfrak{O}_K$ as a $\mathbb{Z}$-module. For positive integers $m\geqslant 1$, write $\mathfrak{M}(\beta_1,\dots,\beta_d,\alpha)$ for the set of maximal elements associated with these generators.

Unboundedness conjecture. For every positive integer $m\geqslant 1$, there exists some $\alpha\in\mathfrak{O}_K^+$ such that

$$\#\mathfrak{M}(\beta_1,\dots,\beta_d,\alpha)\geqslant m.$$

The preceding quadratic-extension result shows that the cardinality of this set can be made arbitrarily large in that case. The conjecture asserts that the same phenomenon holds for arbitrary real number fields.