If $1\le a_1<\cdots<a_n\le 2n$ satisfy $\operatorname{lcm}(a_i,a_j)>2n$ for $i\ne j$, is $a_i>\frac{2n}{3}$ for all $i$?

Question

If integers $1\le a_1<\cdots<a_n\le 2n$ satisfy $\operatorname{lcm}(a_i,a_j)>2n$ for $i\ne j$, is it true that $a_i>\frac{2n}{3}$ for all $i$?

My attempt: Suppose that $i<j$, then $\operatorname{lcm}(a_i,a_j)>a_j$ since if $\operatorname{lcm}(a_i,a_j)=a_j\le 2n$ we have a contradiction. If $\operatorname{lcm}(a_i,a_j)=2a_j$ then $2a_j\ge 3a_i$, and i don't know what to do next.