A finiteness property of postcritically finite unicritical polynomials

Let k be a number field with algebraic closure k, and let S be a finite set of places of k containing all the archimedean ones. Fix d >= 2 and alpha is an element of k such that the map z -> z(d) + alpha is not postcritically finite. Assuming a technical hypothesis on alpha, we prove that there are only finitely many parameters c is an element of k for which z -> z(d) + c is postcritically finite and for which c is S-integral relative to (alpha). That is, in the moduli space of unicritical polynomials of degree d, there are only finitely many PCF k-rational points that are ((alpha), S)-integral. We conjecture that the same statement is true without the technical hypothesis.