The mixed fractional Brownian motion - the sum of independent fractional and standard Brownian motions - is known to be a semimartingale if the Hurst exponent $H$ of its fractional component satisfies $H > 3/4$. The question posed in the title is motivated by recent findings in quantitative finance. In this note, we show that the drift in its Doob-Meyer decomposition has a derivative that is $γ$-Hölder continuous for any $γ< 2H - 3/2$.