The authors continue their investigation of polynomials that are orthogonal with respect to complex oscillatory measures. In this case their focus lies on products of Chebyshev weight functions and highly oscillatory exponential paths localized on the interval \([-1,1]\). They prove that, under certain conditions on the parameters which can be expressed in terms of Bessel functions of the first kind, there exists a sequence of orthogonal polynomials. Finally, the authors give an asymptotic formula for the three-term recursion coefficients.