The magnetic and structural properties of the recently discovered pnictogen/chalcogen-free superconductor LaFeSiH have been investigated by 57Fe synchrotron M & ouml;ssbauer source spectroscopy, X-ray and neutron powder diffraction, and 29Si nuclear magnetic resonance spectroscopy. In contrast with earlier work suggesting the presence of an orthorhombic and magnetic ground state as in underdoped Fe -based pnictides, our results unambiguously establish that LaFeSiH is in fact similar to strongly overdoped Fe -based pnictides: there is no magnetic order (including under hydrostatic pressure up to 18.8 GPa), nor even fluctuating local moments and the system remains tetragonal down to 2 K. This raises the prospect of enhancing the Tc of LaFeSiH by reducing its carrier concentration through appropriate chemical substitutions.