Bubble-up and missing-rim conjecture for wheels of neural codes

Let $\mathcal{C}$ be a neural code, let $\Delta(\mathcal{C})$ be its simplicial complex, and let $\mathcal{W}=(\sigma_1,\sigma_2,\sigma_3,\tau)$ be a wheel of $\mathcal{C}$. A face of $\Delta(\mathcal{C})$ is a max-intersection face if it is the intersection of two or more facets of $\Delta(\mathcal{C})$. Bubble-up and missing-rim conjecture. (i) If $\mathcal{W}$ is a wheel of $\mathcal{C}$, then $\tau\notin\mathcal{C}$. (ii) If $\mathcal{C}$ has a wheel, then it has a wheel $\mathcal{W}=(\sigma_1,\sigma_2,\sigma_3,\tau)$ in which $\tau$ is a max-intersection face of $\Delta(\mathcal{C})$. The preceding results establish these properties for sprockets, wire wheels, and wheel frames; the conjecture asserts both the missing-rim and bubble-up properties for arbitrary wheels.