Denote the space of all bivariate polynomials of total degree not exceeding n by Π_n. The Gasca-Maeztu conjecture [Gasca M. and Maeztu J. I., On Lagrange and Hermite interpolation in R^k, Numer. Math. 39 (1982), 1-14.] states that any Π_n-poised set of nodes, all fundamental polynomials of which are products of linear factors, possesses a maximal line, i.e., a line passing through n+1 nodes. Till now it is proved to be true for n≦ 5. The case n=5 was proved recently in [Hakopian H., Jetter K. and Zimmermann G., The Gasca-Maeztu conjecture for n=5, Numer. Math. 127 (2014), 685-713]. In an earlier paper the following generalized conjecture was proposed by the authors of the present paper: Any Π_n-poised set of nodes, all fundamental polynomials of which are reducible, possesses a maximal curve of some degree k, 1≦ k≦ n-1, i.e., an algebraic curve passing through (1/2)k(2n-k+3) nodes. Clearly the two above conjectures coincide in the case n≦ 2. In this paper we prove that the generalized conjecture is true for n=3.