Convex foliations of degree 4 on the complex projective plane

We show that up to automorphisms of $\mathbb{P}^2_{\mathbb C}$ there are $5$ homogeneous convex foliations of degree four on $\mathbb{P}^2_{\mathbb C}.$ Using this result, we give a partial answer to a question posed in $2013$ by D. {Mar\'in} and J. {Pereira} about the classification of reduced convex foliations on~$\mathbb{P}^2_{\mathbb C}.$