Here is a truly minimalistic and seemingly basic question which should have a simple solution (I hope it does).

Let $A$ be a finite set of integers with the smallest element $0$ and the largest element $l$. The sumset $C:=3A$ resides in the interval $[0,3l]$, and we let $C_1:=C\cap[0,l]$, $C_2:=C\cap[l,2l]$, and $C_3:=C\cap[2l,3l]$.

Is it true that $|C_2|\ge\frac12\,(|C_1|+|C_3|)$, for any choice of the set $A$?

Computations seem to suggest that the answer is in the affirmative.