Prime Hankel representation for the prime-cyclotomic gamma function

Let $\Gamma_{\mathbb P}^{\mathrm{cyc}}$ be the prime-cyclotomic gamma function. A prime Hankel representation should consist of an oriented contour $\mathscr C\subset\mathbb C^\times$, a specified branch of $w^{-z}=\exp(-z\log w)$ along $\mathscr C$, and a canonical prime-cyclotomic kernel $\mathcal K_{\mathbb P}(w)$ independent of $z$, such that on a nonempty vertical strip

$$\frac{1}{\Gamma_{\mathbb P}^{\mathrm{cyc}}(z)}=\frac{1}{2\pi i}\int_{\mathscr C}\mathcal K_{\mathbb P}(w)w^{-z}\,\mathrm dw.$$

The right-hand side should admit entire continuation to $\mathbb C$, with the recurrence $\Gamma_{\mathbb P}^{\mathrm{cyc}}(z+1)=e^{L_{\mathbb P}^{\mathrm{cyc}}(z)}\Gamma_{\mathbb P}^{\mathrm{cyc}}(z)$ and reflection law $\Gamma_{\mathbb P}^{\mathrm{cyc}}(z)\Gamma_{\mathbb P}^{\mathrm{cyc}}(1-z)=1$ induced by explicit transformations of the contour kernel. This would provide a contour analogue of the classical Hankel representation while preserving the defining prime-cyclotomic symmetries; the source does not state whether such a representation is known.