Broadhurst’s dimension conjecture for alternating multiple zeta values

Determine whether the dimension of the \(\Q\)-vector space spanned by alternating multiple zeta values of weight \(N\) satisfies \(\dim_{\Q}\mathcal{Z}^2_N=F_{N+1}\), where \(F_N\) is the Fibonacci sequence defined by \(F_N=F_{N-1}+F_{N-2}\) with \(F_1=F_2=1\).

Background

The paper defines $\mathcal{Z}^2_N$ as the $\Q$-vector space spanned by alternating multiple zeta values of weight $N$. Broadhurst conjectured that its dimension is governed by the Fibonacci sequence.

The text states that Deligne proved the upper bound $\dim_{\Q}\mathcal{Z}^2_N\le F_{N+1}$. The unresolved issue is whether this upper bound is always attained, giving the exact dimension of the space of alternating multiple zeta values in each weight.