Continuity conjecture for the Rényi entropy rate at order one

Let $Z=\{Z_n\}_{n=1}^{\infty}$ be a stationary ergodic process over a finite alphabet $\mathcal{Z}$. Its Shannon entropy rate is

$$H(Z)\triangleq \lim_{n\rightarrow\infty}\frac{H(Z_1^n)}{n},$$

and, when the limit exists, its $\alpha$-th order Rényi entropy rate is

$$H_{\alpha}(Z)\triangleq \lim_{n\rightarrow\infty}\frac{H_{\alpha}(Z_1^n)}{n}.$$

Continuity conjecture. For every stationary ergodic process $Z$,

$$\lim_{\alpha\rightarrow 1}H_{\alpha}(Z)=H(Z).$$

The conjecture is a natural rate-level analogue of the finite-block identity $\lim_{\alpha\rightarrow 1}H_{\alpha}(Z_1^n)=H(Z_1^n)$, but the existence of the Rényi entropy rate itself is not known for general stationary ergodic processes. The source reports this conjecture as neither proved nor disproved in the literature.