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We study injective Jordan semi-triple maps $\\phi:\\mathcal{R}\\to M_n(\\mathbb{D})$, namely injective maps satisfying \\[\n  \\phi(XYX)=\\phi(X)\\phi(Y)\\phi(X), \\qquad \\text{for all } X,Y\\in\\mathcal{R}. \\] Assuming that the centre of $\\mathbb{D}$ has more than two elements, we give a criterion for automatic additivity and show that there are exactly two obstructions. 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We study injective Jordan semi-triple maps $\\phi:\\mathcal{R}\\to M_n(\\mathbb{D})$, namely injective maps satisfying \\[\n  \\phi(XYX)=\\phi(X)\\phi(Y)\\phi(X), \\qquad \\text{for all } X,Y\\in\\mathcal{R}. \\] Assuming that the centre of $\\mathbb{D}$ has more than two elements, we give a criterion for automatic additivity and show that there are exactly two obstructions. 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