The question title says it all. For the record, I have no reason to believe that this is true, but my question has a bit of a background.
I am reading Ramanan's Global Calculus book because I am interested in the isomorphism between singular cohomology with coefficients in a ring $R$ and sheaf cohomology with respect to the constant sheaf $\underline R$ (page 114, Theorem 4.14). He only assumes his topological space $X$ is locally contractible, but he quotes a result (page 8,Lemma 1.14) which assumes his space is hereditarily paracompact ; the result says the sheafification map of a presheaf $\mathcal F(U) \to \mathcal F^+(U)$ is surjective for all open subsets $U \subseteq X$, assuming the existence part of the glueing axiom.
So I was wondering if he just wrote somewhere he would make that assumption for a long time, forgot to assume it, or if the question in my title has a positive answer, which to be honest, would surprise me! But hey, until I have a counter-example... who knows.