Woronowicz-type integral representations for the triple sine function

Let $S_3(z\mid\omega_1,\omega_1,\omega_2)$ denote the triple sine function with two coinciding periods, and let $B_{3,3}(z\mid\omega_1,\omega_1,\omega_2)$ be the triple Bernoulli polynomial. Then the following two integral representations hold:

Woronowicz-type integral representations.

$$\begin{aligned} \log S_3(z\mid\omega_1,\omega_1,\omega_2)={}&-\frac{\pi\mathrm{i}}{6}B_{3,3}(z\mid\omega_1,\omega_1,\omega_2)+\frac{1}{2}\left(1-\frac{z}{\omega_1}\right)\log\left(1-e^{2\pi\mathrm{i}z/\omega_1}\right)\\ &-\frac{1}{2\pi\mathrm{i}}\left(\frac{z-\omega_1}{\omega_2}+\frac{1}{2}\right)\operatorname{Li}_2\left(e^{2\pi\mathrm{i}z/\omega_1}\right)-\frac{\omega_1}{2\pi^2\omega_2}\operatorname{Li}_3\left(e^{2\pi\mathrm{i}z/\omega_1}\right)\\ &+\frac{\omega_2^2}{\omega_1^2}\int_0^\infty\mathrm{d}s\,\log\left(1-e^{-2\pi s}\right)\left[\frac{s+\frac{\mathrm{i}(z-\omega_1)}{\omega_2}}{e^{-\frac{2\pi}{\omega_1}(\omega_2s+\mathrm{i}z)}-1}-\frac{s-\frac{\mathrm{i}(z-\omega_1)}{\omega_2}}{e^{\frac{2\pi}{\omega_1}(\omega_2s-\mathrm{i}z)}-1}\right] \end{aligned}$$

and

$$\begin{aligned} \log S_3(z\mid\omega_1,\omega_1,\omega_2)={}&-\frac{\pi\mathrm{i}}{6}B_{3,3}(z\mid\omega_1,\omega_1,\omega_2)+\frac{\omega_2^2}{4\pi^2\omega_1^2}\operatorname{Li}_3\left(e^{2\pi\mathrm{i}z/\omega_2}\right)-\frac{1}{12}\log\left(1-e^{2\pi\mathrm{i}(z-\omega_1)/\omega_2}\right)\\ &-\frac{\omega_1}{4\pi\omega_2}\int_0^\infty\mathrm{d}s\,\frac{\left(\operatorname{Li}_2\left(e^{-2\pi s}\right)-2\pi s\log\left(1-e^{-2\pi s}\right)\right)\sinh\left(2\pi s\omega_1/\omega_2\right)}{\sinh\left(\frac{\pi}{\omega_2}(\omega_1(s-\mathrm{i})+\mathrm{i}z)\right)\sinh\left(\frac{\pi}{\omega_2}(\omega_1(s+\mathrm{i})-\mathrm{i}z)\right)}. \end{aligned}$$

These formulas are conjectured on the basis of comparisons with Bridgeland's expressions and numerical checks for random values of the arguments; their general validity remains to be proved.