A refinement of an estimate of the arithmetic minimum of the product of nonhomogeneous linear forms (regarding Minkowski's nonhomogeneous conjecture)

One refines an estimate of B. F. Skubenko (Tr. Mat. Inst. Akad. Nauk 148, 218–224 (1978)). Let $\Lambda$ be a point lattice of determinant $d(\Lambda)$ in the $n$-dimensional Euclidean space $\mathbf R^n$, and let $L\in\mathbf R^n$. We consider the nonhomogeneous
$$ M=M(\Lambda,L)=\inf_{(z_1,\dots,z_n)\in\Lambda+L}\prod^n_{i=1}|z_i|. $$
One proves that there exists an effectively computable constant $n_0$ such that if $n\geqslant n_0$, then
$$ M<2^{-\frac n2}e^{20}n^{-\frac37}\log^{\frac47}nd(\Lambda). $$