Context. The interpretation of microlensing results towards the Large Magellanic Cloud (LMC) still remains controversial. White dwarfs have been proposed to explain these results and, hence, to contribute significantly to the mass budget of our Galaxy. However, several constraints on the role played by regular carbon-oxygen white dwarfs exist. Aims. Massive white dwarfs are thought to be made of a mixture of oxygen and neon. Correspondingly, their cooling rate is larger than those of typical carbon-oxygen white dwarfs and they fade to invisibility in short timescales. Consequently, they constitute a good candidate for explaining the microlensing results. Methods. Here, we examine in detail this hypothesis by using the most recent and up...