  Let $X$ be a compact connected strongly pseudoconvex $CR$ manifold of real
dimension $2n-1$ in $\mathbb{C}^{N}$. For $n\ge 3$, Yau solved the complex
Plateau problem of hypersurface type by checking a bunch of Kohn-Rossi
cohomology groups in 1981. In this paper, we generalize Yau's conjecture on
some numerical invariant of every isolated surface singularity defined by Yau
and the author to any dimension and prove that the conjecture is true for local
complete intersection singularities of dimension $n\ge 3$. As a direct
application, we solved complex Plateau problem of hypersurface type for any
dimension $n\ge 3$ by checking only one numerical invariant.
