  Drawing on Vanhove's contributions to mixed Hodge structures for Feynman
integrals in two-di\-men\-sion\-al quantum field theory, we compute two
families of determinants whose entries are Bessel moments. Via explicit
factorizations of certain Wro\'nskian determinants, we verify two recent
conjectures proposed by Broadhurst and Mellit, concerning determinants of
arbitrary sizes. With some extensions to our methods, we also relate two more
determinants of Broadhurst--Mellit to the logarithmic Mahler measures of
certain polynomials.
