Farber–Kaufman–Schütz rationality conjecture for higher topological complexity

Let $X$ be an $(r-1)$-connected finite CW-complex, where $r\geq 2$, and let $TC_n(X)$ denote its higher topological complexity. Define the $TC$-generating function

$$\mathcal{F}_X(x)=\sum_{n=1}^{\infty}TC_{n+1}(X)x^n.$$

Farber–Kaufman–Schütz rationality conjecture. The function $\mathcal{F}_X(x)$ is rational with a single pole of order $2$ at $x=1$; equivalently, there exists $P\in\mathbb{Z}[X]$ such that

$$\mathcal{F}_X(x)=\frac{P(x)}{(1-x)^2}.$$

The paper states that this conjecture is established for the class of finite CW-complexes considered in its abstract, namely those having a unique spherical cohomology class $u\in\widetilde{H}^r(X,\mathbb{Z})$ with $u^k\neq 0$ and dimension $kr$, where $k\geq 2$; the supplied material does not specify the conjecture's status beyond that result.