The possible existence of a liquid-liquid critical point in deeply supercooled water has been a subject of debate in part due to the challenges associated with providing definitive experimental evidence. Pioneering work by Mishima and Stanley [Nature 392, 164 (1998) and Phys.~Rev.~Lett. 85, 334 (2000)] sought to shed light on this problem by studying the melting curves of different ice polymorphs and their metastable continuation in the vicinity of the expected location of the liquid-liquid transition and its associated critical point. Based on the continuous or discontinuous changes in slope of the melting curves, Mishima suggested that the liquid-liquid critical point lies between the melting curves of ice III and ice V. Here, we explore this conjecture using molecular dynamics simulations with a purely-predictive machine learning model based on ab initio quantum-mechanical calculations. We study the melting curves of ices III, IV, V, VI, and XIII using this model and find that the melting lines of all the studied ice polymorphs are supercritical and do not intersect the liquid-liquid transition locus. We also find a pronounced, yet continuous, change in slope of the melting lines upon crossing of the locus of maximum compressibility of the liquid. Finally, we analyze critically the literature in light of our findings, and conclude that the scenario in which melting curves are supercritical is favored by the most recent computational and experimental evidence. Thus, although the preponderance of experimental and computational evidence is consistent with the existence of a second critical point in water, the behavior of the melting lines of ice polymorphs does not provide strong evidence in support of this viewpoint, according to our calculations.