Supremum process of a Cauchy RV

Question

I've asked the same question on stats.stackexchange a week ago to no avail, so here we go again:

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Suppose $X_i$ are $\mathrm{Cauchy}(0,~\gamma)$ IID RV's. Does an expression exist for the CDF of the supremum process of their sum?

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After a literature review (of a matter that is unfortunately beyond my field of study), I've come across a paper by Boyarchenko and Levendorskiĭ that, to the best of my understanding, is the only one that doesn't consider the limiting case and claims the following:

For a Lévy process $X$ and its supremum process $\bar{X}=\sup_{0\leq s \leq t} X_s$, the CDF of the supremum process is given by

$$ \begin{align} P(x+\bar{X}_T \leq a) &= \mathbb{E}\left[\mathbf{1}_{x+\bar{X}_T\leq a}\right]\\ &= 1 - \mathbb{E}\left[\mathbf{1}_{x+\bar{X}_T>a}\right]\\ &= 1 - \frac{1}{2\pi i} \int_{\mathrm{Re}~q = \sigma} \frac{1}{q} \left(\mathcal{E}_q^+\mathbf{1}_{a,~+\infty}\right)(x) \end{align} $$

where $x$ is the initial value of the process $X_T$. They then go on to evaluate the $\mathcal{E}$ term in the integrand as

$$ \left(\mathcal{E}_q^+\mathbf{1}_{a,~+\infty}\right)(x) = \frac{1}{\pi}~\mathrm{Im}\int_{-\infty}^{\infty}e^{i(x-a)\xi}\left( \phi_q^+(\xi)-a_q^+ - \frac{1-a_q^+}{1+i\xi} \right)~\mathrm{d}y $$

after a change variables $\xi = \xi(y)=\exp\left(i\omega_-+y\right)$ where $\omega_-$ is apparently a given and mentioned in one of their earlier papers as well. In my case, I believe $\phi_q^+$ to be given by

$$ \phi_q^+(\xi) = \exp\left( \frac{1}{2\pi i} \int_{ e^{i(-\pi-\omega_-)}\mathbb{R}_+ ~\cup~ e^{i\omega_-}\mathbb{R}_+ } \frac{\xi \ln\left(1 + \psi_\mathrm{st}(\eta)\cdot q^{-1}\right)}{\eta\cdot(\xi-\eta)} \right) $$

where

$$ \psi^\mathrm{st}(\xi) = \xi \cdot (\gamma\pi-i\mu) $$

by virtue of lemmas 4.2 and 2.2 respectively.

So, here are my questions:

1. How to obtain $a_q^+$ in the expression for $\left(\mathcal{E}_q^+\mathbf{1}_{a,~+\infty}\right)(x)$ in the Bromwich integral? Lemmas 4.3 and 4.6 seemingly do not cover the case when $c_+=c_-=\gamma$.

2. The above method relies upon Wiener-Hopf. Why does an independent exponentially distributed variable ($T_q$) with mean $q^{-1}$ need to be introduced? What does $q$ physically represent?

3. Analogously, what does $\omega_-$ represent? The authors only give a loose bound. Does a heuristic exist for its selection?

4. Do you figure a straightforward way exists to condition the supremum process on the path of $X_T$ itself?

I've attempted contacting the authors, again, to no avail as of yet. Thank you very much for any help you can provide.