Near second-order Markov property of MA-filtered first-order Markov processes

Consider a first-order Markov process observed at the output of a length-$N$ moving-average (MA) filter, with output covariance

$$C_{\mathrm{1MP}+\mathrm{MA}}(k;\alpha)=\sum_{j=-N+1}^{N-1}(N-|j|)\sigma^2\,\operatorname{e}^{-\alpha|k-j|}.$$

Near second-order Markov conjecture. The MA filtering of the first-order Markov process generates nearly a second-order Markov process.

This concerns the effective Markov order induced by moving-average filtering of a process with exponentially decaying correlations. The statement is presented as a conjecture, but the supplied text does not specify what “nearly” means or provide evidence resolving it.