Harmse–Remmel conjecture on c-Wilf and strong c-Wilf equivalence

Let $P,Q \in\mathcal F_{n,k}$, where $\mathcal F_{n,k}$ denotes the relevant family of generalized Euler permutations. They are c-Wilf equivalent when $A_P(0,t)=A_Q(0,t)$, and strongly c-Wilf equivalent when $A_P(x,t)=A_Q(x,t)$, where

$$A_P(x,t)=\sum_{n\geq0}\frac{t^n}{(kn)!}\sum_{F\in\mathcal F_{n,k}}x^{P\text{-mch}(F)}.$$

Harmse–Remmel conjecture. $P$ and $Q$ are c-Wilf equivalent if and only if they are strongly c-Wilf equivalent. The conjecture extends Nakamura's $k=1$ formulation to $k\geq2$; the source does not state whether it has been resolved.