On the successive minima of the extended logarithmic height of algebraic numbers

Suppose that $\mathbb K\subseteq\mathbb C$ is an algebraic field; $S=2$ if $\mathbb K$ is complex, and $S=1$ if $\mathbb K\subseteq\mathbb R$; $\delta=[\mathbb K:\mathbb Q]/S$. For $\alpha\in\mathbb K^*$ let $H_*(\alpha)=\max\bigl\{\delta h(\alpha),|\ln\alpha|\bigr\}$, where $h(\alpha)$ is the Weil height of the number $\alpha$. Then the inequality
$$ H_*(\alpha_1)\dotsb H_*(\alpha_n)2.5^n(e^{0.2n}n)^S\delta\ln(4.64\delta)>1 $$
holds for multiplicatively independent $\alpha_1,\dots,\alpha_n\in\mathbb K^*$.