Proving convergence of sum over $\mathbb{Z}^n$

The sum diverges already for $n=4$. To see this, let $L\geq K$ be a dyadic parameter, and consider the contribution of $q_1\asymp L^{1/8}$, $q_2\asymp L^{1/6}$, $q_3\asymp L^{1/4}$, $q_4\asymp L^{1/2}$. If the implied constants are sufficiently close to each other, these ranges are pairwise disjoint. However the contribution of such a range is $\asymp L^{1/8+1/6+1/4+1/2}/L>1$. Summing up over the various $L$'s, the claim follows.

The series indeed diverges for $n\ge4$. Indeed, without loss of generality $K\ge1$. For real $A\ge1$ and $p>1$ we have
\begin{equation} \int_1^\infty\frac{dq}{A+q^p}\ge\int_1^\infty\frac{dq}{(A^{1/p}+q)^p} \gg\frac1{A^{1-1/p}}. \end{equation} So, for $n\ge4$ \begin{align*} S_K&\ge\int_{[1,\infty)^n} \frac{dq_1\cdots dq_n}{K+q^{2n}_1+\cdots+q^{8}_{n-3}+q^{6}_{n-2}+q^4_{n-1}+q^2_n} \\ &\gg\,\int_{[1,\infty)^{n-1}} \frac{dq_1\cdots dq_{n-1}}{K+q^{n}_1+\cdots+q^{4}_{n-3}+q^{3}_{n-2}+q^2_{n-1}} \\ &\gg\,\int_{[1,\infty)^{n-2}} \frac{dq_1\cdots dq_{n-2}}{K+q^{n/2}_1+\cdots+q^{2}_{n-3}+q^{3/2}_{n-2}} \\ &\gg\,\int_{[1,\infty)^{n-3}} \frac{dq_1\cdots dq_{n-3}}{K+q^{n/6}_1+\cdots+q^{2/3}_{n-3}} =\infty. \end{align*}

The authors just posted a correction to the paper on arXiv, where they say:

"In an earlier version of this paper we used $2n − 2j + 2$ as the exponent in the definition of BK instead of $4n − 2j + 2.$ As a result (as was pointed out to us) the series SK might not converge. However, the only properties we use of BK are that SK converges and that the ratio of the value of BK at certain points limits to 1."