The generating function for words avoiding soft streaks

Let $n,k,r\in\mathbb{N}_0$ with $k\geq 2$. A soft streak of length $k$ is a non-decreasing word of length $k$, and the generalized binomial coefficient $\operatorname{B}_s(n,k,r)$ is the coefficient of $x^r$ in $(1+x+x^2+\cdots+x^{k-1})^n$. Let $\psi_{k,r}$ and $\omega_k$ denote the quantities used in the Goulden–Jackson generating-function framework, and define

$$f(z)=\frac{(1-z^k)^n}{\displaystyle\sum_{r=0}^{(k-1)n}\psi_{k,r}\operatorname{B}_s(n,k,r)z^r}.$$

The generating-function conjecture. The generating function $f(z)$ is given by the displayed expression above.

This conjectural formula is intended to count words avoiding non-decreasing subwords of length $k$ and to connect the generalized binomial coefficients with the Goulden–Jackson method. The supplied excerpt does not state whether the conjecture has been proved or refuted.