In a remark to Green’s conjecture, Paranjape and Ramanan analysed the vector bundle E which is the pullback by the canonical map of the universal quotient bundle T_Pg−1(−1) on P^g−1 and stated a more general conjecture and proved it for the curves with Clifford Index 1 (trigonal and plane quintics). In this paper, we state the conjecture for general linear systems and obtain results for the case of hyper-elliptic curves.