This question is a generalization of this old, unanswered question.
Suppose we're given a strong PRP $E:\mathcal K\times\mathcal M\to\mathcal C,(K,M)\mapsto C=E(K,M)$. Suppose further we pick a constant parameter $S\in\mathcal M$.
Is $F_S:\mathcal K\to \mathcal C, X\mapsto E(X,S)$ a secure one-way-function?
If the above is not answerable: Is $F_S$ at least a secure OWF for practical purposes?
As example instances one can imagine $S=0^{128},E=\operatorname{AES}$ or $E$ being a fixed-key AES instance within an Even-Mansour construction.