Completeness conjecture for the $S^n$ functional

For a stratified $n$-manifold $(S^n,\bigcup_i S_i)$, let ${S_i}_{i\text{ in }I}$ be transversely intersecting, unoriented PL surfaces, with the stated stratification by their intersections, and suppose every $(n+1)$-fold intersection is empty. If $e_i$ is the Euler number of $S_i$, define

$$Z(S^n,\bigcup_i S_i)=\prod_{i=1}^n 2^{1-\frac{e_i}{4}}.$$

Completeness conjecture for the $S^n$ functional. The $S^n$ functional $Z$ is complete positive and complete finite. The conjecture occurs in the paper's discussion of extending the methods to all dimensions; no resolution is supplied.