Consider the Clausen function $\operatorname{Cl}_2(x)$ that can be defined for $0<x<2\pi$ in several equivalent ways:
$$\begin{align}\operatorname{Cl}_2(x)&=-\int_0^x\ln\left(2\sin\left(\tfrac t2\right)\right)dt\\&=\sum_{n=1}^\infty\frac{\sin\,(n x)}{n^2}\\&=\Im\operatorname{Li}_2\left(e^{i x}\right)\\&=i\left(\frac{\pi^2}6+\frac{x^2}4-\frac{\pi x}2-\operatorname{Li}_2\left(e^{i x}\right)\right).\end{align}\tag1$$ I'm interested in integrals of the form $$I(p)=\int_0^{2\pi}\operatorname{Cl}_2(x)^2\,x^p\,dx.\tag2$$ I found that $$I(0)=\frac{\pi^5}{90}\tag3$$ and conjectured next several values based on numeric evidence: $$I(1)\stackrel?=\frac{\pi^6}{90},\ \ I(2)\stackrel?=\frac{44\,\pi^7}{2835},\ \ I(3)\stackrel?=\frac{23\,\pi^8}{945}.\tag4$$ One might expect that $I(4)$ is a rational multiple of $\pi^9$, but apparently it is not (unless the denominator is huge).
I'm asking for your help in proving conjectured values $(4)$, finding a closed form of $I(4)$, and, if possible, a general formula for $I(p)$.
Update: Values $I(-1)$ and $I(-2)$ are also interesting.