Sun's conjecture on the constants $s_1(n)$ and $s_2(n)$

Let $n>1$ and $r>0$ be integers. Define $s_r(n)$ as the least positive integer $k$ such that every sequence of $k$ integer vectors in the relevant rank-$r$ setting contains a selection of $n$ vectors whose sum is divisible by $n$ but not divisible by $n^2$, as in the paper's definition. Sun's conjecture. For every integer $n>2$,

$$s_1(n)=2n+1\qquad\text{and}\qquad s_2(n)=4n+1.$$

The source derives lower bounds matching these proposed values in the relevant cases and poses the equalities based on computation; it supplies no proof of the conjecture.