A special case of maximal rank conjecture states that for a general curve $C$ and general points $p_1,\dots ,p_n\in C$ the map $$Sym^2H^0(K_C-p_1-\dots -p_n)\to H^0(K_C^{\otimes 2}-2p_1-\dots -2p_n)$$ has maximal rank, i.e. is either injective or surjective. (This special case is verified (by Ballico and Ellia using degeneration and also by Payne and Jensen using tropical methods) and it is a theorem now)

I need the same result where I let all the points coalesce, that is, I want that for a general $C$ and general point $p\in C$ the map $$Sym^2H^0(K_C-n.p)\to H^0(K_C^{\otimes 2}-2n.p)$$ has maximal rank. I thought one could be able to prove this by reducing it to the first problem, but I failed miserably after hours of trying :)

Can anybody throw in some ideas?