Asymptotics of Polynomials Orthogonal on a Cross with a Jacobi-type Weight

We investigate asymptotic behavior of polynomials $Q_n(z)$ satisfying non-Hermitian orthogonality relations $$\int_\Delta s^kQ_n(s)\rho(s)\dd s =0, \quad k\in\{0,\ldots,n-1\},$$ where $\Delta := [-a,a]\cup [-\ic b,\ic b]$, $a,b>0$, and $\rho(s)$ is a Jacobi-type weight.