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We investigate if the active galactic nucleus (AGN) of Mrk 590, whose supermassive black hole was until recently highly accreting, is turning off due to a lack of central gas to fuel it.We analyse new subarcsecond resolution Atacama Large Millimetre/submilllimetre Array maps of the 12CO(3-2) line and 344 GHz continuum emission in Mrk 590.We detect no 12CO(3-2) emission in the inner 150 pc, constraining the central molecular gas mass to M(H2) ≲ 1.6 × 105M☉, no more than a typical giant molecular gas cloud, for a CO luminosity to gas mass conversion factor of αCO ~ 0.8M☉ (K km s-1 pc2)-1. However, there is still potentially enough gas to fuel the black hole for another 2.6 × 105 yr assuming Eddington-limited accretion. We therefore cannot rule out that the AGN may just be experiencing a temporary feeding break, and may turn on again in the near future. We discover a ring-like structure at a radius of ~1 kpc, where a gas clump exhibiting disturbed kinematics and located just ~200 pc west of the AGN, may be refuelling the centre. Mrk 590 does not have significantly less gas than other nearby AGN host galaxies at kpc scales, confirming that gas reservoirs at these scales provide no direct indication of on-going AGN activity and accretion rates. Continuum emission detected in the central 150 pc likely originates from warm AGN-heated dust, although contributions from synchrotron and free-free emission cannot be ruled out. [ABSTRACT FROM AUTHOR]