Restricted van der Waerden theorem for nilprogressions

For $k\geq 2$, let $\sum_{<k}$ be the collection of words in $*_1,\ldots,*_d$ in which each letter occurs at most $k-1$ times. For elements $a,x_1,\ldots,x_d$ of an $s$-step nilpotent group $G$, a nilprogression of step $s$, length $k$ and rank $d$ is

$$A=\left\{w(x_1,\ldots,x_d)a:w\in\sum_{<k+1}\right\}.$$

It is non-degenerated when $|A|=|\sum_{<k+1}|$. A set is monochromatic for a finite partition when it is contained in one class. Restricted van der Waerden theorem for nilprogressions. For every $k\geq 1$ and $d\geq 2$, there exists a $k$-step nilpotent group $(G,\cdot)$ with $d$ generators and a set $V\subset G$ such that $V$ contains no non-degenerated nilprogression of step $k$, length $k+1$ and rank $d$, but every finite partition of $V$ has a class containing a non-degenerated nilprogression of step $k$, length $k$ and rank $d$. This is a nilpotent-group analogue of the restricted van der Waerden theorem, extending the question from arithmetic progressions to nilprogressions. The source attributes the conjecture to Johnson Jr. and Richter; no resolution is supplied in the given material.