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This article presents doubly differential measurements of the asymmetries in production rates between mesons containing a charm quark and those containing an anti-charm quark in proton-proton collisions at a centre-of-mass energy of s=13.6 TeV using data recorded by the LHCb experiment. The asymmetries of D0, D+ and Ds+ mesons are measured for two-dimensional intervals in transverse momentum and pseudorapidity, within the range 2.5 < pT< 25.0 GeV/c and 2.0 < η < 4.5. No significant production asymmetries are observed. Comparisons to the Pythia 8 and Herwig 7 event generators are also presented, and their agreement with the data is evaluated. These measurements constitute the first measurements of production asymmetries at this centre-of-mass energy of colliding beams, and the first measurements with the LHCb Run 3 detector.