The maximal transcendence-degree conjecture for double zeta values of equal parity

Let $a$ and $b$ be different positive integers of the same parity, with $b\geqslant 2$. Let $N=a+b$, and let $I(a,b)$ and $J(a,b)$ be the sets introduced in the source. Consider the set

$$\{2\pi i,\zeta(k),\zeta(a,b)\}_{k\in (I(a,b)\cup J(a,b))\setminus\{0,N\}}.$$

Maximal transcendence-degree conjecture. The transcendence degree of the $\mathbb Q$-extension field generated by this set is maximal.

The paper proves that this assertion would follow from the period conjecture by comparing the fraction fields associated with the period matrix. The maximality assertion itself is not proved.