The simplified exponents-region conjecture for K-hop networks

Let $K\geq 2$, let $R_1,\ldots,R_K$ be the link rates, and let $\epsilon_1,\ldots,\epsilon_K$ be the type-I error thresholds at the $K$ decision centers. Let $\mathcal{E}^*(R_1,\ldots,R_K,\epsilon_1,\ldots,\epsilon_K)$ denote the fundamental type-II error-exponents region, and choose a permutation $\pi\colon\{1,\ldots,K\}\to\{1,\ldots,K\}$ such that

$$\epsilon_{\pi(1)}\geq\epsilon_{\pi(2)}\geq\cdots\geq\epsilon_{\pi(K)},$$

with $\epsilon_{\pi(0)}:=1$. For $i\in\{1,\ldots,K\}$, define

$$\ell_i^*:=\max_{\ell}\{\ell\colon\ell\in\{\pi(i),\ldots,\pi(K)\}\}.$$

Simplified exponents-region conjecture. The region $\mathcal{E}^*(R_1,\ldots,R_K,\epsilon_1,\ldots,\epsilon_K)$ is the set of all exponent tuples $(\theta_1,\ldots,\theta_K)$ for which there exist nonnegative rates $R_{i,\ell}$ satisfying

$$\theta_k\leq\min_{i\in\{1,\ldots,\pi(k)\}}\left[\sum_{\ell=1}^{k}\eta_\ell(R_{i,\ell})\right],\qquad k\in\{1,\ldots,K\},$$

and

$$R_\ell\geq\sum_{\substack{i\in\{1,\ldots,K\}:\\ \ell_i^*\geq\ell}}\left(\epsilon_{\pi(i-1)}-\epsilon_{\pi(i)}\right)R_{i,\ell},\qquad \ell\in\{1,\ldots,K\}.$$

The formula is obtained by retaining only the rate-sharing components indexed by the ordered decision centers; it is proved in the paper for $K=2$ and $K=3$, whereas its validity for arbitrary $K\geq2$ is conjectured.