KöMaL 2017 A712

We say that a strictly increasing positive real sequence $a_1,a_2,\dots$ is an elf sequence if for any $c>0$ we can find an $N$ such that $a_n<cn$ for $n=N,N+1,\dots\,.$ Furthermore, we say that $a_n$ is a hat if $a_{n-i}+a_{n+i}<2a_n$ for $1\le i\le n-1$. Is it true that every elf sequence has infinitely many hats?