Linear stability for line bundles over curves

Let C be a smooth irreducible projective curve and let (L, H^0(L)) be a complete and globally generated linear series on C. Denote by M_L the syzygy bundle, kernel of the evaluation map H^0(L) ⊗ O_C → L. In this work we restrict our attention to the case of globally generated line bundles L over a curve with h0(L) = 3. The purpose of this short note is to connect Mistretta-Stoppino Conjecture on the equivalence between linear (semi)stability of L and slope (semi)stability of M_L with the existence of extensions of line bundles of L by certain quotients Q of M_L. Also, we give numerical conditions to produce examples of line bundles L which are linearly semistables but with syzygy bundle M_L unstable, that is, we find numerical conditions to look for counter-examples to Mistretta-Stoppino Conjecture of rank 2.