No, because all your hypersurface may be contained in the locus where the conclusion of Voisin's result does not hold.

Voisin's paper concerns varieties of general type, Ulmer's preprint is partly an extension of Voisin's strategy to hypersurfaces of degree 6k in P(1,1,2k,3k), and for your problem it would be probably sufficient to extend Voisin's method to the family of hypersurfaces of degree 2d in Ps(1,1,1,d) with d^2 nodes.