Capacity characterization for the Poisson channel with increasing power

Consider the Poisson channel

$$x[t] \geq 0 \quad \longrightarrow \quad y[t] \sim \operatorname{Pois}(x[t]),$$

with power constraint $\tfrac{1}{n}\sum_{t=1}^n x[t] \leq n^\alpha$. This is the Poisson channel with increasing power (PCIP). A normalized rate $\rho>0$ is achievable if, for every $\epsilon>0$, one can design a code for some blocklength $n$ with error probability at most $\epsilon$ and rate $\rho\log n$; let $\bar C(\alpha)$ be the closure of achievable normalized rates. PCIP capacity conjecture. For the PCIP, $\bar C(\alpha)=\alpha/2$. This characterizes the maximum achievable normalized rate when the power constraint grows polynomially with the blocklength; the supplied text does not establish the claim's resolution.