  As an application of the residue functions corresponding to the lc-measures
developed by the authors, the proof of the injectivity theorem on compact
K\"ahler manifolds for plt pairs by Matsumura is improved in this article to
allow multiplier ideal sheaves of plurisubharmonic functions with neat analytic
singularities in the coefficients of the relevant cohomology groups. With the
use of a refined version of analytic adjoint ideal sheaves, a plan towards a
solution to the generalised version of Fujino's conjecture (i.e. an injectivity
theorem on compact K\"ahler manifolds for lc pairs with multiplier ideal
sheaves) is laid down and, in addition to the result for plt pairs, a proof for
lc pairs in dimension 2, which is also an improvement of Matsumura's result, is
given.
