Question

Let $S$ be a fixed finite set of valuations on $\mathbb Q$ containing the archimedean one. A $S$-unit is $x\in\mathbb Q$ such that $|x|_v =1$ for all $v\notin S$. For any $S$-units $x_0, \dots, x_n$ we let $$ N=N(x_0,\dots, x_n)= \prod_{v\in S} |x_0+\cdots + x_n|_v $$ and $$ H=H(x_0,\dots, x_n)= \prod_{v\in S} \max \{ |x_0|_v, \dots, |x_n|_v\}\enspace. $$ It is known that whenever subsums of $x_0+\cdots+x_n$ do not vanish, then for any $\epsilon>0$, we have $$N>cH^{1-\epsilon}$$ for some $c>0$ depending only on $\epsilon$ and $n$ ($S$ is fixed, so I don't keep track of this dependence). I would like to know more precisely if there exists a lower bound with an explicit dependence on $n$. In particular, I would like to know if there is a lower bound of the form $$ N> C^{-n^k} H^{1-\delta} $$ for some absolute constants $C,k,\delta>0$.