Multiple series connected to Hoffman's conjecture on multiple zeta values

Recent results of Zlobin and Cresson-Fischler-Rivoal allow one to decompose any suitable $p$-uple series of hypergeometric type into a linear combination (over the rationals) of multiple zeta values of depth at most $p$; in some cases, only the multiple zeta values with 2's and 3's are involved (as in Hoffman's conjecture). In this text, we study the depth $p$ part of this linear combination, namely the contribution of the multiple zeta values of depth exactly $p$. We prove that it satisfies some symmetry property as soon as the $p$-uple series does, and make some conjectures on the depth $p-1$ part of the linear combination when $p=3$. Our result generalizes the property that (very) well-poised univariate hypergeometric series involve only zeta values of a given parity, which is crucial in the proof by Rivoal and Ball-Rivoal that $\zeta(2n+1)$ is irrational for infinitely many $n \geq 1$.