Linear independence conjecture for non-commutative polynomial classes

Let $p\neq 2$. Let $A={\mathbb Z}\{S\}$ be a non-commutative polynomial ring over a set $S$, and let $X(A)$ be the group containing the classes $\langle f\rangle$ for elements $f\in A$. Let $\{f_i\}_{i=1}^r$ be a finite set of distinct non-zero elements of $A$ such that $f_i\neq -f_j$ for every $i\neq j$.

Linear independence conjecture. The subset $\{\langle f_i\rangle\}_{i=1}^r$ of $X(A)$ is $\mathbb Z$-linearly independent.

This conjecture concerns the non-commutative polynomials underlying the construction of universal pre-Witt and Witt functors. The paper states that its validity implies that the functor $E$ is universal as a pre-Witt functor, and that the associated functor $\hat{E}$ is universal as a Witt functor when $p\neq 2$. Its resolution is not specified in the supplied text.