because any nontrivial lower bound would provide some nontrivial irrational/transcendental results which is related to a variant of Grothendieck’s period conjecture (see [@Del]). For example, we can easily show that $\MPV(2,4)=\langle \log^2 2, \pi^2, \pi \log 2\sqrt{-1},
(K-1)\sqrt{-1}\rangle,$ where $K=\sum_{n\ge 0} (-1)^n/(2n+1)^2$ is the Catalan’s constant. From Grothendieck’s conjecture we know $d(2,4)=4$ (see op. cit.) but we don’t have a unconditional proof yet. On the other hand, we may obtain upper bound of $d(w,N)$ by finding as many linear relations in $\MPV(w,N)$ as possible. As in the cases of MZVs and the alternating Euler sums the regularized double shuffle relations (RDS) play important roles in revealing the relations among MPVs. We shall study this theory for MPVs in section \[sec:RDS\] by generalizing some results of [@IKZ] (also cf. [@Bigotte]). It is commonly believed that in levels one and two all linear relations among MPVs are consequences of RDS.

From the point of view of Lyndon words and quasi-symmetric functions Bigotte et al. [@Bigotte] have studied MPVs (they call them *colored MZVs*) primarily by using double shuffle relations. However, when the level $N\ge 3$, these relations are not complete in general, as we shall see in this paper.

If the level 