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Reconstruction techniques for intrinsic quasar continua are crucial for the precision study of Lyman-$α$ (Ly-$α$) and Lyman-$β$ (Ly-$β$) transmission at $z>5.0$, where the $λ<1215 A$ emission of quasars is nearly completely absorbed. While the number and quality of spectroscopic observations has become theoretically sufficient to quantify Ly-$α$ transmission at $5.0<z<6.0$ to better than $1\%$, the biases and uncertainties arising from predicting the unabsorbed continuum are not known to the same level. In this paper, we systematically evaluate eight reconstruction techniques on a unified testing sample of $2.7<z<3.5$ quasars drawn from eBOSS. The methods include power-law extrapolation, stacking of neighbours, and six variants of Principal Component Analysis (PCA) using direct projection, fitting of components, or neural networks to perform weight mapping. We find that power-law reconstructions and the PCA with fewest components and smallest training sample display the largest biases in the Ly-$α$ forest ($-9.58\%/+8.22\%$ respectively). Power-law extrapolations have larger scatters than previously assumed of $+13.1\%/-13.2\%$ over Ly-$α$ and $+19.9\%/-20.1\%$ over Ly-$β$. We present two new PCAs which achieve the best current accuracies of $9\%$ for Ly-$α$ and $17\%$ for Ly-$β$. We apply the eight techniques after accounting for wavelength-dependent biases and scatter to a sample $19$ quasars at $z>5.7$ with IR X-Shooter spectroscopy, obtaining well-characterised measurements for the mean flux transmission at $4.7<z<6.3$. Our results demonstrate the importance of testing and, when relevant, training, continuum reconstruction techniques in a systematic way.