  In 2010, Brasselet, Sch\"urmann and Yokura conjectured an equality of
characteristic classes of singular varieties between the Goresky-MacPherson
$L$-class $L_*(X)$ and the Hirzebruch homology class $T_{1*}(X)$ for a compact
complex algebraic variety $X$ that is a rational homology manifold. In this
note we give a proof of this conjecture for projective varieties based on
cubical hyperresolutions, the Decomposition Theorem, and Hodge theory. The
crucial step of the proof is a new characterization of rational homology
manifolds in terms of cubical hyperresolutions which we find of independent
interest.
