  We generalize Koll\'ar's conjecture (including torsion freeness, injectivity
theorem, vanishing theorem and decomposition theorem) to Saito's $S$-sheaves
twisted by a $\mathbb{Q}$-divisor. This gives a uniform treatment for various
kinds of Koll\'ar's package in different topics in complex geometry. As a
consequence we prove Koll\'ar's package of pluricanonical bundles twisted by a
certain multiplier ideal sheaf. The method of the present paper is
$L^2$-theoretic.
