General cyclic-system conjecture for contextuality

Let $\left(\mathfrak{S},\mathfrak{C}\right)$ be a system with

$$\mathfrak{S}=\left\{\left(V_i,W_{\pi(i)}\right):i\in\{1,\ldots,n\}\right\},$$

where $\pi$ is a circular permutation of $\{1,\ldots,n\}$, and

$$\mathfrak{C}=\left\{\left(V_i,W_i\right):i\in\{1,\ldots,n\}\right\}.$$

Here $\Delta_0(\mathfrak{C})$ is the sum of minimal coupling discrepancies and $\Delta_{\min}(\mathfrak{S},\mathfrak{C})$ is its minimum over couplings. General cyclic-system conjecture. The minimum discrepancy and degree of contextuality are

$$\Delta_0\left(\mathfrak{C}\right)=\frac{1}{2}\sum_{i=1}^{n}\left|\left\langle V_i\right\rangle-\left\langle W_i\right\rangle\right|,$$ $$\Delta_{\min}\left(\mathfrak{S},\mathfrak{C}\right)=\frac{1}{2}\max\left(2\Delta_0\left(\mathfrak{C}\right),\;\mathsf{s}_{odd}\left(\left\langle V_iW_{\pi(i)}\right\rangle:i\in\{1,\ldots,n\}\right)-n+2\right),$$

and

$$\mathsf{CNTX}\left(\mathfrak{S},\mathfrak{C}\right)=\frac{1}{2}\max\left(0,\mathsf{s}_{odd}\left(\left\langle V_iW_{\pi(i)}\right\rangle:i\in\{1,\ldots,n\}\right)-n+2-\sum_{i=1}^{n}\left|\left\langle V_i\right\rangle-\left\langle W_i\right\rangle\right|\right).$$

Consequently, the system has a noncontextual description if and only if

$$\mathsf{s}_{odd}\left(\left\langle V_iW_{\pi(i)}\right\rangle:i\in\{1,\ldots,n\}\right)\le n-2+\sum_{i=1}^{n}\left|\left\langle V_i\right\rangle-\left\langle W_i\right\rangle\right|.$$

The conjecture unifies the SZLG, EPRB, and KCBS systems as cyclic systems of orders $3$, $4$, and $5$. The source notes that a proof was available by February 2015, with a supporting lemma proved in the cited reference; a later passage states the corresponding noncontextuality criterion as a theorem.