Formula for alternating double t-values with two odd indices

Derive a formula expressing the alternating double t-values t(\overline{2a+1},\overline{2b+1}) in terms of products of Riemann zeta values.

Background

The paper establishes unramifiedness results and explicit evaluations for several families of alternating multiple t-values. However, the authors report that they could not find an expression for the alternating double t-values with both indices odd and barred, namely t(\overline{2a+1},\overline{2b+1}), solely in terms of products of Riemann zeta values.

The remark notes that the motivic Galois descent criterion does not determine the depth of the classical multiple zeta values appearing in such expressions, because the coradical and depth filtrations do not agree. Numerical evidence further suggests that depth-two and depth-four multiple zeta values may be required in general, leaving the requested zeta-product formula unresolved.