The complement symmetry conjecture for Varchenko's path sums

Let $\lambda$ be a partition in an $(n-k)\times k$ rectangle, let $\lambda^c$ be its complement, and let $\alpha$ be the automorphism of $\mathbb{C}(\hbar,a_1,\ldots,a_n)$ defined by $\alpha(a_i)=-a_{n+1-i}$ for $1\leq i\leq n$, with $\alpha(\hbar)=\hbar$. Complement symmetry conjecture.

$$\mathbb{V}(\lambda)=\alpha\bigl(\mathbb{V}(\lambda^c)\bigr).$$

The source later states that this fails at the level of multisets, so the conjecture is disproved even though there is a natural bijection between paths to $\lambda$ and paths to $\lambda^c$.