Parker's improved conjecture on type-I complementary arrays

Let $n$ be a positive integer. A type-I complementary array over the alphabet ${0,1,-1}$ is an array represented by a function $f_{n-1,0}({\bf{x}})$, with the indices and parameters $s$, $q$, $p$, and $w$ as in the construction preceding the claim. Parker's improved conjecture. For any $n$, each type-I complementary array over the alphabet ${0,1,-1}$ is of the form

$$f_{n-1,0}({\bf{x}}) = \left(\prod_{j=0}^{|s|-1} \left(x_{s(j)} + x_{q(s(j))} + w({s(j)}) + 1\right)\right)(-1)^{\sum_{j=0}^{|p|-1} w({p(j)})x_{p(j)} + \sum_{j=0}^{|p|-2}x_{p(j)}x_{p(j+1)}}.$$

This claim strengthens the earlier conjecture cited in the source by characterising all type-I complementary arrays through the displayed construction; its resolution is not established in the supplied text.