Let $D : SU(2)\mapsto\mathbb{B}(\mathbb{C}^{d})$ be an unitary irreducible representation of SU(2). Denote the spin of this representation by S (i.e. $d = 2S+1$).
Define the functional $F$ by
$$F(\psi) = \int_{SU(2)} {\left\lvert{\left\langle \psi\right|}D(g){\left|\psi\right\rangle }\right\rvert}^{4}\,dg , \phantom{xx} \psi \in \mathbb{C}^{d} , \phantom{x} \|\psi\| = 1 ,$$
where $dg$ is the Haar measure of SU(2).
Question : Is $F$ maximized by the coherent states of $D$ ?
Remarks :
1) For $S = 1$ the coherent states are maximizer, but there are also other maximizer.
2) For $S = 3/2$ the coherent states are the only maximizer.
3) We can write $F$ as
$$F(\psi) = \sum_{j=0}^{2S} \frac{{(Tr((P\otimes P)P_{j}))}^{2}}{Tr P_{j}} = \sum_{j=0}^{2S} \frac{{(Tr((P\otimes P)P_{j}))}^{2}}{2j+1}$$
where $P$ is the orthonormal projection on the span of $\psi$ and $P_{j}$ is the orthonormal projection on the states with total spin $j$.
Note that $Tr((P\otimes P)P_{j}) = 0$ if $2S - j$ is odd.