Functional equations of polygonal type for multiple polylogarithms in weights $5$, $6$ and $7$

We present new functional equations in weights 5, 6 and 7 use them for explicit depth reduction of multiple polylogarithms. These identities generalize the crucial identity $\mathbf{Q}_4$ from recent work Goncharov Rudenko that was used their proof weight 4 case Zagier's Polylogarithm Conjecture.