In previous work, we considered a filtering and parameter estimation strategy in a rough volatility model for high-frequency data based on a Markovian representation of fractional (resp. Liouville) Brownian motion. However, frameworks based on processes of this type are not suitable for capturing other stylized facts associated with volatility (such as long memory): they are not stationary and, because of the self-similarity property, they cannot account for both roughness and long-term persistence. In this work, in the spirit of Bennedsen et al. (2022), we devise an alternative model that is rich enough to accommodate both effects, while preserving a structure suitable for similar filtering and estimation strategies. We consider a setup where observations are modeled as a doubly-stochastic Poisson process, while the state process can be approximated with a finite-dimensional superposition of Ornstein-Uhlenbeck processes and is constructed in such a way that sample paths exhibit roughness whereas the autocorrelation function decays polynomially. This framework allows for a more flexible parametrization and provides a foundation for a satisfactory and thorough empirical analysis of real tick-by-tick data. In particular, given their interpretation and role in the model, we provide different estimation strategies for different parameters: while those governing the long-term behavior of signal and observations can be inferred via moment methods, particle-based methods are more suitable for those controlling the roughness of sample paths (essentially as they are very sensitive to the unobservability of the volatility process).