  In this notes we reprove MacPherson's conjecture on $L^2-(n,q)$-cohomology
through Demailly's formulation of H\"ormander's Estimate. This approach allows
us to weaken the condition of locally semipositivity in Ruppenthal's
$L^2$-representation of adjoint bundle. Moreover we prove the MacPherson's
conjecture of twisted coefficient bundle under an arbitrary singular hermitian
metric. As applications, we study the MacPherson type problem for
pluri-canonical bundle.
