Towards a Jucys-Murphy theory for the Okada algebras

I'll discuss work in progress which aims to construct Jucys-Murphy elements in the Okada algebra $\mathcal{F}_n$ together with a corresponding notion of content for the Young-Fibonacci lattice which encodes the spectrum of the Jucys-Murphy elements with respect to the Fibonacci-Tableau bases for irreducible $\mathcal{F}_n$-representations.