The scarcity of white dwarf-brown dwarf binaries in the solar neighbourhood: A population synthesis study

White dwarf--brown dwarf (WD--BD) binaries are intrinsically rare systems that provide a unique opportunity to study binary evolution and the formation of substellar companions. Despite recent observational progress, only a small number are known in the solar neighbourhood. We investigate whether their scarcity can be explained by binary formation and evolution together with observational selection effects. We extend the MRBIN binary population synthesis code into the substellar regime by incorporating brown dwarf evolutionary models and an extended initial mass function. We simulate the stellar population within 100 pc of the Sun and compare the predicted WD--BD population with the observed sample. Our simulations predict 13+-5 WD--BD systems within 100 pc, consistent with observations. These systems represent only 0.1% of the local white dwarf population, confirming their rarity. The models reproduce the overall distributions of white dwarf masses, temperatures, and orbital periods, although the simulated brown dwarfs are systematically hotter than observed. Close WD--BD binaries are underproduced with the standard common-envelope efficiency 0.3, suggesting that higher efficiencies may be required to reproduce the observed short-period systems. The simulations also reproduce the observed deficit of systems at intermediate orbital periods between post-common-envelope binaries and wide non-interacting systems, as well as a weakened brown dwarf desert in main-sequence--brown dwarf binaries.