Exact sequences of $\mathcal{O}_X$-modules, sections over X minus a point, and splitting

Question

Let $X$ be a (let's say irreducible) scheme, let $x$ be a closed point, put $U = X - \{x\}$. Let $0\to\mathcal{F}\to\mathcal{G}\to\mathcal{H}\to 0$ be an exact sequence of $\mathcal{O}_X$-modules. If the exact sequence of sections over $U$ is split, is the original sequence of $\mathcal{O}_X$-modules split? The case I am most interested in is when $X$ is projective.