In this paper, we study the Pythagoras number \({\mathcal {P}}({\mathcal {O}}_K)\) for the rings of integers in totally real biquadratic fields K. We continue Tinková’s work toward proving the conjecture of Krásenský, Raška and Sgallová that a biquadratic K satisfies \({\mathcal {P}}({\mathcal {O}}_K)\ge 6\) if and only if it contains neither \(\sqrt{2}\) nor \(\sqrt{5}\) , with only finitely many exceptions. We fully resolve two of the three remaining classes of fields by proving that all but finitely many fields K containing \(\sqrt{6}\) or \(\sqrt{7}\) satisfy \({\mathcal {P}}({\mathcal {O}}_K)\ge 6\) . Furthermore, we present ideas and computations that further support the conjecture for fields K containing \(\sqrt{3}\) . This enables us to refine the conjecture by explicitly listing the exceptional fields.
