Remote and webcam-based eye tracking in multi-line reading suffers from various noise factors and layout ambiguity, precisely where real-time reading support needs reliable, per-fixation line assignment. Prior work largely addresses this challenge post hoc or by restricting behavior (e.g., disallowing re-reading), undermining interactive use. We propose CONF-LA (Confidence-score-based Online Fixation-to-Line Assignment), a principled, low-latency approach that integrates knowledge about reading behavior and Gaussian line likelihoods over fixations to compute a posterior-line-score and defers assignments when uncertainty is high. Evaluated on existing open-source data, CONF-LA demonstrates stable performance in post hoc analysis and closes the online-offline gap (1-2 %) with a mean per-fixation latency of 0.348 ms. Our approach exhibits particular invariance toward regressions, yielding significant improvement in ad hoc median accuracies on children data (approx. 95 %) over all tested algorithms. We encourage further research in this direction and discuss possibilities for future development.