Let $\phi$ be a real function on unit disc $D$ in $\mathbb C$. Assume $\Delta \phi\le 0$. Then there is a curve $\gamma$ from the center of $D$ such that $$\int\limits_\gamma e^\phi<\infty.$$

The question came from my failed answer to this question. I know that the answer is YES, but I do not see a direct proof.