Let $(M,g,\sigma)$ be a compact Riemmannian surface equipped with a spin structure $\sigma$. For any metric $\tilde{g}$ on $M$, we denote by $\mu_1(\tilde{g})$ (resp. $\lambda_1(\tilde{g})$) the first positive eigenvalue of the Laplacian (resp. the Dirac operator) with respect to the metric $\tilde{g}$. In this paper, we show that $$\inf \frac{\lambda_1(\tilde{g})^2 }{\mu_1(\tilde{g})} \leqslant \frac{1}{2}.$$ where the infimum is taken over the metrics $\tilde{g}$ conformal to $g$. This answer a question asked by Agricola, Ammann and Friedrich
