On non-symmetric $t$-convexity

Zsolt P'ales asked whether there exists a function $f:\mathbb{R}\to \mathbb{R}$ such that $$f\left(\frac{x+2y}{3}\right)\le \frac{f(x)+2f(y)}{3}\text{ for all }x\le y$$ which is not midconvex. We show that the answer is negative.