Relative Schinzel–Zassenhaus conjecture for generalized house

Fix a positive integer $m$. For an algebraic integer $\alpha\in\overline{\mathbb{Q}}$, define

$$\operatorname{house}_m(\alpha):=\max_{[\mathbb{Q}(\alpha):K]\leq m,\,\sigma:K\hookrightarrow\mathbb{C}}M_{\sigma(K)}(\alpha),$$

where $M_{\sigma(K)}(\alpha)$ is the relevant relative Mahler measure; for $m=1$, this specializes to the ordinary house. Relative Schinzel–Zassenhaus conjecture. There is an absolute constant $c>0$ such that, for every nonzero algebraic integer $\alpha$ that is not a root of unity,

$$\log\operatorname{house}_m(\alpha)>\frac{c}{[\mathbb{Q}(\alpha):\mathbb{Q}]}.$$

This is proposed as a relative-height analogue of the Schinzel–Zassenhaus conjecture, following the resolved weaker bound of Dimitrov; its status remains open.