Sun's conjecture on the new constant $w_r(n)$

Let $n>1$ and $r>0$ be integers. Define $w_r(n)$ as the least positive integer $k$ such that, for any $k$ integer vectors ${\bf a}_i=(a_{i1},\ldots,a_{ir})$ not congruent to ${\bf 0}=(0,\ldots,0)$ modulo $n$, there is an $I\subseteq\{1,\ldots,k\}$ such that

$$\sum_{i\in I}{\bf a}_i\equiv{\bf 0}\pmod n$$

but

$$\sum_{i\in I}{\bf a}_i\not\equiv{\bf 0}\pmod {n^2}.$$

Sun's conjecture. Let $n>1$ be an integer. (i) For any $2n-1$ integers $a_1,\ldots,a_{2n-1}$ not divisible by $n$, there is an $I\subseteq\{1,\ldots,2n-1\}$ with $|I|\in\{n,n+1\}$ such that $\sum_{i\in I}a_i$ is divisible by $n$ but not divisible by $n^2$. (ii) We have $w_2(n)=4n-3$. These claims are motivated by the preceding exact results for $w_1(n)$ and by the equality $s(\mathbb Z_n^2)=4n-3$; the source reports them as conjectures based on computation, with no resolution supplied.