An $n$-dimensional (closed) pseudomanifold is a finite simplicial complex $X$ such that
(i) every simplex is a face of an $n$-simplex
(ii) every $(n-1)$-simplex is a face of exactly two $n$-simplices
(iii) Given any two $n$-simplices $\sigma, \tau \in X$ there is a sequence of $n$-simplices $\sigma_0 = \sigma, \ldots, \sigma_k = \tau$ such that $\sigma_i \cap \sigma_{i+1}$ is an $(n-1)$-dimensional simplex for each $0 \leq i \leq k-1$.
These conditions imply that the polyhedron of a pseudomanifold is path-connected. Is it true that if a finite simplicial complex $X$ satisfies (i) and (ii) and has a path-connected polyhedron then it satisfies (iii)?
Counterexample with $n=2$: let $X$ be the union of two hollow tetrahedra joined at a vertex. Here (i) and (ii) hold; (iii) fails, yet $X$ is path-connected.