A Dimca-Greuel type inequality for foliations

Let F be a holomorphic foliation at p∈ C², and B be a separatrix of F. We prove the following Dimca-Greuel type inequality μₚ(F,B)/τₚ(F,B)<4/3, where μₚ(F,B) is the multiplicity of F along B and τₚ(F,B) is the dimension of the quotient of C[[x,y]] by the ideal generated by the components of any $1$-form defining F and any equation of B. As a consequence, we provide a new proof of the 4/3-Dimca-Greuel's conjecture for singularities of irreducible plane curve germs, with foliations ingredients, that differs from those given by Alberich-Carrami\~nana, Almir\'on, Blanco, Melle-Hern\'andez and Genzmer-Hernandes but it is in line with the idea developed by Wang.