In this paper we conjecture that the piecewise linear map f(x) = px for 0 ≤ x < 1/p, f(x) = sx - s/p for 1/p ≤ x ≤ 1, p > 1, 0 < s < 1, which has an expanding, onto branch and a contracting branch, is eventually piecewise expanding. We give a partial proof of the conjecture, in particular for values of p and s such that ⌈− ln(p(1−s)+s)/lns⌉ ≠ ⌈−lnp/lns⌉.