How do I find a pattern with several wildcard strings in a suffix tree?

Question

Let's say that I got a suffix tree ST(T) and a pattern S₁ * S₂ ... * S_k, whereas * represents any string out of the same alphabet as the strings S₁,...,S_k.

How do I find the pattern using the indexes of the leaf nodes?

My idea for 2 strings S₁ and S₂ would be to put every occurence of S₁ and S₂ into integer lists a₁ and a₂ and check if any integer in a₂ is greater or equal than any integer in a₁ + s₁.length.

Am I on the right track? And is it a similar method for k strings?