A generalization of Fisher's inequality (Q1279662)

Let \(L\) be a collection of \(k\) positive integers and \(F\) a family of subsets of \(\{1,\dots,n\}\) such that \(| X\cap Y|\in L\) for all \(X,Y\in F\) with \(X\neq Y\). The author conjectures that then \[ | F|= \sum^k_{i=0} {n-1\choose i}. \] He proves this conjecture for collections \(L\) satisfying some additional condition. Very recently he proved the conjecture for any \(L\).