Injectivity of the universal regularized double shuffle map and isomorphism with the µ-MHZV algebra

Determine whether the Q[µ]-algebra of µ-multiple Hurwitz zeta values Z^µ is isomorphic to the universal algebra R_RDS defined by the regularized double shuffle relations; equivalently, prove that the canonical maps (ϕ∗, ϕ⊔) from the universal object (R_RDS,·) to any target algebra satisfying the regularized double shuffle relations are injective. Concretely, establish whether (Z^µ,·) ≅ (R_RDS,·).

Background

Section 6 introduces the universal algebra (R_RDS,·) characterized by the regularized double shuffle (RDS) relations for µ-multiple Hurwitz zeta values (µ-MHZVs). For any Q[µ]-algebra (R,·) equipped with maps satisfying the finite double shuffle relations, there exist unique comparison maps (ϕ∗, ϕ⊔) from (R_RDS,·) to (R,·) making the relevant diagrams commute. The central structural question is whether these universal relations capture all algebraic relations among µ-MHZVs.

The problem asks if the canonical maps are injective, equivalently whether the algebra generated by µ-MHZVs is exactly the universal RDS algebra. This mirrors the celebrated conjecture of Ihara–Kaneko–Zagier for classical multiple zeta values (level one). The authors note that at level one (µ=1, integer shifts) they expect an affirmative answer, while at level two (µ=2) the situation is subtler because double shuffle relations are known not to generate all Q-linear relations among Euler sums; nonetheless, µ-MHZVs differ from Euler sums, leaving the level-two case unresolved.

References

The following conjecture describes the combinatorial structure of the algebra of µ-MHZVs. Problem 6.4. Is the map (ϕ∗,ϕ⊔) always injective? Equivalently, is the algebra (Zµ,·) of µ-MHZVs isomorphic to (R_RDS,·)? When µ = 1 and m’s are positive integers, we expect the answer to be affirmative since this is essentially the level one case where we have the well-known conjecture by Ihara, Kaneko, and Zagier [9]. But when µ = 2 and m’s are positive integers, we are at level two and we know the double shuffle relations do not generate all Q-linear relations among Euler sums (see the remark after [16, Theorem 1.1]. However, there is some subtle difference between µ-MHZVs and the Euler sums so we do not know the answer to the problem in this case.

— Regularized double shuffle relations of $μ$-multiple Hurwitz zeta values  (2402.11689 - Li et al., 2024) in Problem 6.4, Section 6: The universal algebra and a conjecture