I asked a question at M.SE a couple of years ago about polylogarithms $\!^{[1]}$$\!^{[2]}$$\!^{[3]}$$\!^{[4]}$ where I conjectured $$720\,\operatorname{Li}_4\!\left(\tfrac12\right)-2160\,\operatorname{Li}_4\!\left(\tfrac13\right)+2160\,\operatorname{Li}_4\!\left(\tfrac23\right)+270\,\operatorname{Li}_4\!\left(\tfrac14\right)+540\,\operatorname{Li}_4\!\left(\tfrac34\right)+135\,\operatorname{Li}_4\!\left(\tfrac19\right)\\ =19\pi^4+30\left(\pi^2-\beta^2\right)\left(10\alpha^2-12\alpha\beta+3\beta^2\right)-30\,\alpha^2\left(19\alpha^2-24\alpha\beta+8\beta^2\right)\!,\\ \text{where $\alpha=\ln2,\,\,\beta=\ln3$.}$$ I put a bounty on it recently, but it still remains unanswered. So, I decided to re-post it here. Also, I recently discovered several similar conjectures for $\operatorname{Li}_5(z)$ involving powers of the golden ratio $\phi=\frac{1+\sqrt5}2$: $$4455\,\operatorname{Li}_5\left(\phi^{-2}\right)-1215\,\operatorname{Li}_5\left(\phi^{-4}\right)-360\,\operatorname{Li}_5\left(\phi^{-6}\right)+15\,\operatorname{Li}_5\left(\phi^{-12}\right)\\ =3015\,\zeta(5)-702\ln^5\phi+180\,\pi^2\ln^3\phi-38\,\pi^4\ln\phi$$

$$45000\,\operatorname{Li}_5\left(\phi^{-2}\right)-16875\,\operatorname{Li}_5\left(\phi^{-4}\right)-144\,\operatorname{Li}_5\left(\phi^{-10}\right)+9\,\operatorname{Li}_5\left(\phi^{-20}\right)\\ =28944\,\zeta(5)-6000\ln^5\phi+1600\,\pi^2\ln^3\phi-356\,\pi^4\ln\phi$$

$$15660\,\operatorname{Li}_5\left(\phi^{-2}\right)-19440\,\operatorname{Li}_5\left(\phi^{-4}\right)+7680\,\operatorname{Li}_5\left(\phi^{-6}\right)+2430\,\operatorname{Li}_5\left(\phi^{-8}\right)-15\,\operatorname{Li}_5\left(\phi^{-24}\right)=5025\,\zeta(5)+2088\ln^5\phi-240\,\pi^2\ln^3\phi-28\,\pi^4\ln\phi$$

How can we prove these conjectures? Are there other similar identities of this kind?