Upper bound on the maximal length of a 2-increasing sequence of triples

For each positive integer $n$, let $[n]$ denote the set $\{1, 2, \ldots, n\}$. For triples $a = (a_1, a_2, a_3)$ and $b = (b_1, b_2, b_3)$ of integers, we say that $a$ is $2$-less than $b$, denoted $a <_2 b$, if $a_i < b_i$ for at least two coordinates $i \in \{1, 2, 3\}$. A sequence of triples is called $2$-increasing if every pair of distinct elements in the sequence satisfies the $2$-less relation in the order they appear. Let $F(n)$ denote the maximal length of a $2$-increasing sequence of triples with each coordinate belonging to $[n]$. Prove that for every natural number $n$,

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