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A family f₁,..., fₙ of operators on a complete metric space X is called contractive if there exists a positive λ < 1 such that for any x,y in X we have $d(f_i\left(x\right),f_i\left(y\right))\le \lambda d(x,y)$ for some i. Austin conjectured that any commuting contractive family of operators has a common fixed point, and he proved this for the case of two operators. We show that Austin’s conjecture is true for three operators, provided that λ is sufficiently small.