Integral-basis conjecture for Euler sums

For a positive integer $n$, an Euler sum of weight $n$ is a signed multiple zeta value whose absolute indices sum to $n$.

Integral-basis conjecture. There exist $\mathbb{Q}$-linearly independent Euler sums of weight $n$ such that every Euler sum of weight $n$ is a $\mathbb{Z}$-linear combination of these sums.

This conjecture proposes an integral basis in each weight, strengthening the preceding question about linear bases over $\mathbb{Q}$. The source gives no resolution status.