Lang's conjecture on lower bounds for linear forms in logarithms

Let $\epsilon>0$. For rational positive numbers $a_1,\dots,a_n$ and integers $b_1,\dots,b_n$, define

$$B_j=\max\{|b_j|,1\},\qquad A_j=\max\{\mathrm e^{\mathrm h(a_j)},1\},\qquad B=\max_{1\le j\le n}B_j.$$

Assume that $b_1\log a_1+\cdots+b_n\log a_n\ne0$. Lang's conjecture. There exists a constant $C(\epsilon)>0$ depending only on $\epsilon$ such that

$$|b_1\log a_1+\cdots+b_n\log a_n|>\frac{C(\epsilon)^nB}{(B_1\cdots B_nA_1^2\cdots A_n^2)^{1+\epsilon}}.$$

This is a conjectural lower bound for nonzero linear forms in logarithms of rational numbers. In the paper it is discussed as an input that would yield stronger estimates for values of binomial forms; its resolution status is not established in the supplied text.