Ramaekers's strong $n$-conjecture

Let $n\geq 3$. For an $n$-tuple $a=(a_1,\ldots,a_n)\in\mathbb{Z}^n$ with nonzero entries, write

$$q(a)=\frac{\log(\max\{|a_1|,\ldots,|a_n|\})}{\log\operatorname{rad}(a_1\cdots a_n)},$$

and for a sequence $A=\{a^{(1)},a^{(2)},\ldots\}$ of such tuples write $Q_A=\limsup_{k\to\infty}q(a^{(k)})$. Let $R(n)$ be the set of $n$-tuples $(a_1,\ldots,a_n)\in\mathbb{Z}^n$ satisfying the stated sum, subsum, and pairwise-coprimality conditions. Ramaekers's conjecture. For every $n\geq 3$, $Q_{R(n)}=1$.

This is a strong generalisation of the $abc$-conjecture. The paper's abstract states that the conjecture is false for every $n\geq 5$, while the case $n=3$ is equivalent to the $abc$-conjecture.