We will relate the complex Plateau problem to some new numerical invariant of isolated singularities and Shokurov’s minimal discrepancy conjecture. As a result, we will show that this invariant is strictly positive for every isolated 3 dimensional canonical Gorenstein singularity and higher dimensional local complete intersection singularity. In particular, the vanishing of this invariant gives the interior regularity of the Harvey-Lawson solution to the complex Plateau problem.