While speech quality is typically assessed on complete utterances, streaming and generative systems require incremental estimation from partial audio. Existing predictors assume full context, degrading on prefix-constrained inputs. Extending ARECHO, we propose ANCHOR, reformulating incremental assessment as a multi-resolution autoregressive task. It models chunk- and utterance-level quality within a single decoder using dual-resolution tokens and a resolution-aware hierarchy for coarse-to-fine refinement. Experiments show substantial robustness under partial input, including a 48% PLCMOS error reduction on 2-second prefixes. Convergence analysis reveals a 4-6 s effective perceptual context horizon. A stress test further isolates structured extrapolation biases under localized corruption. Results demonstrate that hierarchical supervision improves incremental prediction and elucidates how perceptual quality accumulates over time.