Conjectured distribution formula for the ascending Lévy process supremum

Let $X_t$ be a Lévy process whose Lévy–Khintchine exponent $f$ is a balanced Rogers function. Let $\zeta_f$ be the canonical parametrisation of $\gamma_f$, let $\lambda_f(r)=f(\zeta_f(r))$, let $f_{[\zeta]}^\uparrow$ be the Wiener–Hopf factor, and let $F_{f\uparrow}(r;x)$ be the function described by the preceding definitions. Assume

$$\sup\{\operatorname{Im}\zeta_f(r):r\in(0,\infty)\}<\infty,$$ $$\sup\{\vartheta_{f\uparrow}(r):r\in(0,\infty)\}<\frac{\pi}{2},$$

and, for some $t_0\in[0,\infty)$,

$$\int_1^\infty \frac{|f_{[\zeta_f(r)]}^\uparrow(i\overline{\zeta_f(r)})|}{f_{[\zeta_f(r)]}^\uparrow(0^+)}e^{-t_0\lambda_f(r)}\lambda_f'(r)\,dr<\infty.$$

Conjectured supremum-distribution formula. For $t>t_0$ and $x>0$,

$$\mathbf{P}(X_t^\uparrow<x)=\int_0^\infty\frac{|f_{[\zeta_f(r)]}^\uparrow(i\overline{\zeta_f(r)})|}{f_{[\zeta_f(r)]}^\uparrow(0^+)}F_{f\uparrow}(r;x)e^{-t\lambda_f(r)}\lambda_f'(r)\,dr.$$

This would give a semi-explicit expression for the distribution of the ascending process supremum. The formula is expected to follow by interchanging integrations in the Laplace-transform representation and then inverting the transform; the required Fubini justification is not established here. The boundedness assumption on $\operatorname{Im}\zeta_f(r)$ may also be difficult to relax, even for strictly stable Lévy processes.