Finite-family Hermite matrix conjecture for cubic irrational pairs

Let $\triangle$ be the domain of pairs of real numbers used by the multidimensional continued fraction algorithms, and let a finite family of such algorithms determine a corresponding Hermite matrix whose rows are the associated sequences of non-negative integers. A pair $(\alpha,\beta)\in\triangle$ is in the same cubic number field when $\alpha$ and $\beta$ belong to a common cubic number field.

Finite-family Hermite matrix conjecture. There is a family of multidimensional continued fractions spanned by finitely many multidimensional continued fraction algorithms such that

$$(\alpha,\beta)\text{ are in the same cubic number field}$$

if and only if some row of the corresponding Hermite matrix is eventually periodic.

This is presented as a reformulation of a conjecture attributed to Karpenkov and as a proposed best-possible form of the Hermite problem: periodicity in a suitable finite family of algorithms should characterize pairs of reals lying in the same cubic number field. The supplied text does not state whether the conjecture has been resolved.