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We derive the quantitative modulus of continuity which we conjecture to be optimal for solutions of the p-degenerate two-phase Stefan problem. Even in the classical case p = 2, this represents a twofold improvement with respect to the early 1980's state-of-the-art results by Caffarelli- Evans (Arch Rational Mech Anal 81(3):199-220, ) and DiBenedetto (Ann Mat Pura Appl 103(4):131-176, ), in the sense that we discard one logarithm iteration and obtain an explicit value for the exponent α( n, p). [ABSTRACT FROM AUTHOR]