The Brown measure of unbounded variables with free semicircular imaginary part

Let $x_0$ be a possibly-unbounded self-adjoint random variable, $tildesigma_alpha$ and $sigma_beta$ are semicircular variables with variances $alphageq 0$ and $beta>0$ respectively (when $alpha = 0$, $tildesigma_alpha = 0$). Suppose $x_0$, $sigma_alpha$, and $tildesigma_beta$ are all freely independent. We compute the Brown measure of $x_0+tildesigma_alpha+isigma_beta$, extending the previous computations which only work for bounded self-adjoint random variable $x_0$. The Brown measure in this unbounded case has the same structure as in the bounded case; it has connections to the free convolution $x_0+sigma_{alpha+beta}$. We also compute the example where $x_0$ is Cauchy-distributed.