A simplicial polyhedral fan $\Sigma \subseteq \mathbb{R}^n$ is a collection of simplicial polyhedral cones $\{\alpha_1 v_1 + \dots + \alpha_r v_r : \alpha_r \geq 0\} \subseteq \mathbb{R}^n$, where the $v_i$ are linearly independent, such that faces of cones are contained in $\Sigma$ and two cones intersect on faces. It is complete if the cones of $\Sigma$ span of all $\mathbb{R}^n$. Assume furthermore that each cone $\sigma$ is strongly convex, $\sigma \cap -{\sigma} = \{0 \}$.
This defines a simplicial complex $\Delta(\Sigma)$ which has a vertex for each dimension 1 cone and the vertices span a simplex if their corresponding cones span a higher-dimensional cone. A simplicial complex is shellable if there exists a well-order $\tau_1 \prec \dots \prec \tau_t$ on the maximal cones of $\Sigma$ such that the simplicial complex $$\Delta(\tau_j \cap \bigcup_{i < j} \tau_i)$$ satisfies that every cone in it is contained in a maximal cone (of codimension 1 in $\Sigma$), which is also called pure. My question is whether this simplicial complex is always shellable? In my opinion its geometric realisation should be a polytope, which are shellable as is shown in Ziegler's book "Lectures on Polytopes". His notion is of shellability is stronger than ours but should imply ours.
On the other hand the article of Kleinschmidt and Onn states that this is an open problem to which the answer is likely negative. The examples of non-shellable simplicial complexes I could find, all use this stronger notion of shellability, for example Rudin's ball, which does not imply non-shellability in our sense.