The Browkin–Brzeziński strong $n$-conjecture for subsum-free integer tuples

For a non-zero integer $m$, let $\operatorname{rad}(m)$ be its largest square-free positive divisor. For $n\geq 3$ and $a=(a_1,\ldots,a_n)\in\mathbb{Z}^n$ with all $a_i\neq 0$, define

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

For a sequence $A=\{a^{(1)},a^{(2)},\ldots\}\subseteq\mathbb{Z}^n$, define

$$Q_A=\limsup_{k\rightarrow\infty}q(a^{(k)}).$$

Let $A(n)\subseteq\mathbb{Z}^n$ be the set of $n$-tuples satisfying (Z) $a_1+\cdots+a_n=0$, (S1) no nontrivial subsum with coefficients in $\{0,1\}$ vanishes, and (G1) $\operatorname{gcd}(a_1,\ldots,a_n)=1$. The Browkin–Brzeziński $n$-conjecture. For every $n\geq 3$, one has

$$Q_{A(n)}=2n-5.$$

This is one of the strong $n$-conjectures studied over the rational integers; the supplied material does not state whether it has been resolved.