Denote $[n]=\{1,2,\dots,n\}$. Assume $n\geq2$.

Question. Is it true that given any $S_1,S_2,\dots,S_{2n}$ (repetition allowed) subsets of $[2n]$ with $a\in S_a$ and $\# S_a=a$ for all $1\leq a\leq 2n$, there exist $i, j, k\in[2n]$ (not all equal) such that $$i\in S_j, \qquad j\in S_k, \qquad k\in S_i \,\, ?$$