Au's conjecture on iterated integrals of unital functions

Let $\mathbf{k}\in\mathbb{N}^d$, and let $\mathsf{U}_N$ denote the set of $N$-unital functions. For $U(x)\in\mathsf{U}_N$, write $\operatorname{Li}_{\mathbf{k}}(U(x))$ for the corresponding multiple polylogarithm, and let $\mathsf{CMZV}_{w}^N$ denote the space of colored multiple zeta values of weight $w$. The integral below is understood in its renormalized sense.

Au's conjecture. For all $U(x)\in\mathsf{U}_N$, the renormalized value

$$\int_0^1 \frac{1}{x}\operatorname{Li}_{\mathbf{k}}(U(x))\,dx$$

belongs to $\mathsf{CMZV}_{|\mathbf{k}|+1}^N$.

This conjecture is intended to provide a crucial mechanism for evaluating Apéry-like sums through iterated integrals involving unital functions. The supplied text attributes it to Au but gives no resolution or further evidence, so its status remains open.