An explicit theory of $\pi_{1}^{\un,\crys}(\mathbb{P}^{1} - \{0,\mu_{N},\infty\})$ - II-2 : From standard algebraic relations of weighted multiple harmonic sums to those of cyclotomic $p$-adic multiple zeta values

Let $X_{0}=\mathbb{P}^{1} - (\{0,\infty\} \cup \mu_{N})\text{ }/\text{ }\mathbb{F}_{q}$, with $N \in \mathbb{N}^{\ast}$ and $\mathbb{F}_{q}$ of characteristic $p>0$ and containing a primitive $N$-th root of unity. We establish an explicit theory of the crystalline pro-unipotent fundamental groupoid of $X_{0}$. In part I, we have computed explicitly the Frobenius, and in particular cyclotomic $p$-adic multiple zeta values. In part II, we use part I to understand the algebraic relations of cyclotomic $p$-adic multiple zeta values via explicit formulas ; this is in particular a study of the harmonic Ihara actions and the maps of comparisons between them introduced in I-2 and I-3. In II-1, we have developed the basics of algebraic theory of cyclotomic sequences of prime weighted multiple harmonic sums and adjoint cyclotomic multiple zeta values viewed as variants of those of the algebraic theory of cyclotomic multiple zeta values. In this II-2, we use part I and II-1 to show that one can read some standard algebraic relations of cyclotomic $p$-adic multiple zeta values via the explicit formulas and via the standard algebraic relations of sequences of multiple harmonic sums. This amounts to say that the harmonic Ihara actions and the comparison maps are compatible with algebraic relations. The two main results are two "harmonic" versions of Besser-Furusho-Jafari's theorem that $p$-adic multiple zeta values satisfy the regularized double shuffle relations. This gives two different answers to what we could call "the adjoint variant" of a question of Deligne and Goncharov about reading the quasi-shuffle relation of $p$-adic multiple zeta values via explicit formulas.