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For every $p \\in (1,\\infty)$, we prove that the closed $\\mathrm{L}^p$-realization $\\mathcal{D}_{E,p}$ of the Dolbeault-Dirac operator is bisectorial and admits a bounded $\\mathrm{H}^\\infty$ functional calculus on $\\mathrm{L}^p(\\Omega^{0,\\bullet}(M,E))$. 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