Convexity conjecture for degraded CMLOBC capacity regions

Let $q=2$, $m=3$, and $l=2$, and let the degraded CMLOBCs have channel matrices

$$\mathbf{S}^{(i)}=\bigl(\epsilon_{0}^{(i)}\mathbf{J}_{7\times1}\mid\epsilon_{1}^{(i)}\mathbf{S}_{21}\mid\epsilon_{2}^{(i)}\mathbf{I}_{7\times7}\bigr),$$

with $\mathbf{S}_{21}$ a stochastic incidence matrix of two-dimensional versus one-dimensional subspaces of $\mathbb{F}_{2}^{3}$. The channel is degraded under the conditions $\epsilon_{0}^{(1)}\leq\epsilon_{0}^{(2)}$ and $\epsilon_{0}^{(1)}+\epsilon_{1}^{(1)}\leq\epsilon_{0}^{(2)}+\epsilon_{1}^{(2)}$.

Convexity conjecture. For the degraded CMLOBCs described above, the capacity region is strictly concave ($\cap$) if and only if

$$\epsilon_{1}^{(1)}\epsilon_{2}^{(2)}>\epsilon_{1}^{(2)}\epsilon_{2}^{(1)}.$$

The preceding examples suggest that superposition coding gives no benefit over time sharing outside Case (i), but the source says that proving the conjecture in full generality is difficult. No resolution is given.