## 1 Introduction  
In high-bandwidth networks, coherent optical communication systems play a crucial role in addressing the ever-growing demand for data transmission capacity. However, these systems face significant challenges, particularly in managing phase noise that affects coherent receivers. Pilot-assisted techniques for tracking phase noise in optical systems may be essential in current low-margin optical network scenarios, where maintaining signal integrity is crucial. Despite notable progress in developing robust compensation techniques, as highlighted in [1, 2], phase noise remains a substantial barrier, especially as data rates and modulation complexities increase. The minimum distance estimator (MDE), for instance, has demonstrated superior capabilities in tracking phase noise across various quadrature amplitude modulation (QAM) formats. However, its complexity increases with the modulation order, requiring a larger number of trial phases and longer filter lengths for denser constellations, thereby imposing heavier computational burdens. This increase in processing may affect communication latency. Optimizing resource efficiency is crucial, particularly for high-order modulation schemes that are resource intensive. In green optical networks, for example, power management, especially through power scaling and adaptive modulation, is vital. Therefore, exploring alternative strategies or innovative combinations of the MDE and other phase-tracking methods is essential. In the design of low-margin optical networks, the system margin (S-margin) accounts for time-varying network conditions, including slow-varying impairments due to network aging. This aging process can cause laser de-tuning, leading to misalignment with optical filters in intermediate nodes, such as reconfigurable optical add-drop multiplexers (ROADMs). Typically, transponder aging introduces a margin of 0.5 dB [3]. These mismatches in local oscillator lasers can affect the laser linewidth, which directly increases phase noise, negatively impacting overall system performance. Advancements in this field aim to achieve an optimal balance between improving system performance and reducing resource consumption, thereby ensuring the scalability and reliability of future optical communication networks. Recent studies have emphasized the potential of pilot-aided carrier phase recovery (PA-CPR) as a promising approach to address phase random-walk challenges in coherent optical receivers [4; 5; 6; 7; 8; 9]. The effectiveness of PA-CPR in mitigating phase noise is particularly noteworthy, as pilot symbols experience the same phase perturbations as data symbols, providing a reliable reference for phase correction. Typically, these pilot symbols are high-energy, known deterministic symbols such as binary phase shift keying (BPSK) or quadrature phase shift keying (QPSK), with well-defined phase angles, enabling accurate phase estimation. This approach can facilitate the development of new algorithms for low-margin network systems. The adoption of new technologies, such as hollow core fibers (HCFs), has expanded to various applications, including low-latency transmission, data centers, radio front-haul, and experimental data transmission using coherent and direct detection transceivers [10]. These advances present significant opportunities for dense wavelength division multiplexing (DWDM) networks in the coming years. HCFs offer substantial advantages over traditional single-mode fibers (SMFs) and photonic shield core fibers (PSCFs), such as extremely low optical nonlinearity (over 1000 times weaker than SMFs), higher optical power handling, longer repeater spacing, reduced attenuation, new low-loss transmission bands, and low chromatic dispersion, making them highly suitable for very-high-order modulation formats. Consequently, phase estimation algorithms for these technologies should be able to efficiently operate up to 256 QAM and higher-order modulation formats.