Multi-process maximum-age threshold probing policy for age-of-information minimization

Let $N>1$, let $E$ denote the energy state, let $T_1,\ldots,T_N$ denote the ages of information of the $N$ processes, and define

$$\bm{T}_{-k}=[T_1,T_2,\ldots,T_{k-1},T_{k+1},\ldots,T_N].$$

Let $E_p$ and $E_s$ be the probing and sampling energy costs, respectively. For $E\geq E_p+E_s$, write $T_{th}(E,\bm{T}_{-k})$ for a threshold depending on $E$ and the ages other than $T_k$. Multi-process threshold-policy conjecture. The optimal probing policy for the $\alpha$-discounted age-of-information cost minimization problem is a threshold policy on the largest process age: the optimal action is to probe the channel state if and only if

$$\arg\max_{1\leq k\leq N}T_k\geq T_{th}(E,\bm{T}_{-k}).$$

This is the proposed policy structure for the multi-process remote-sensing problem with energy harvesting; the supplied text does not state whether the claim has been proved or remains open.