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Context. Excess microwave emission, commonly known as anomalous microwave emission (AME), is now routinely detected in the Milky Way. Although its link with the rotation of interstellar (carbonaceous) nano-grains seems to be relatively well established at cloud scales, large-scale observations show a lack of correlation between the different tracers of nano-carbons and AME, which has led the community to question the viability of this link. Aims. Using ancillary data and spinning dust models for nano-carbons and nano-silicates, we explore the extent to which the AME that come out of the Galactic Plane might originate with one or another carrier. Methods. In contrast to previous large-scale studies, our method is not built on comparing the correlations of the different dust tracers with each other, but rather on comparing the poor correlations predicted by the models with observed correlations. This is based on estimates that are as realistic as possible of the gas ionisation state and grain charge as a function of the local radiation field and gas density. Results. First, nano-carbon dust can explain all the observations for medium properties, in agreement with the latest findings about the separation of cold and warm neutral medium in the diffuse interstellar medium. The dispersion in the observations can be accounted for with little variations in the dust size distribution, abundance, or electric dipole moment. Second, regardless of the properties and abundance of the nano-silicate dust we considered, spinning nano-silicates are excluded as the sole source of the AME. Third, the best agreement with the observations is obtained when the emission of spinning nano-carbons alone is taken into account. However, a marginal participation of nano-silicates in AME production cannot be excluded as long as their abundance does not exceed $Y_{\rm Si} \sim 1\%$.