Equality in Hausdorff–Young for hypergroups

It has been shown in "On the Hausdorff-Young theorem for commutative hypergroups" by Sina Degenfeld-Schonburg, that one can extend domain of Fourier transform a hypergroup $K$ to $L^p(K)$ $1\leq p \leq 2$, and inequality holds true these cases. In this article, we examine structure non-zero functions which equality is attained inequality, $1<p<2$, further provide characterization basic uncertainty principle hypergroups with non-trivial centre.