The fundamentalness conjecture for special unipotent representations of unitary groups

Let $U(p,q)$ be a real unitary group, and let a special unipotent representation mean a representation in the special-unipotent class considered in the paper. A representation is fundamental when its associated combinatorial $\theta$-stable datum is fundamental, equivalently in the $U(p,q)$ setting when all neighbouring $\lambda_a$-blocks have differences at most $1$. Fundamentalness conjecture. Every special unipotent representation of $U(p,q)$ is fundamental. This would place all special unipotent representations inside the proposed core of the unitary dual; the supplied text presents the assertion as a conjecture and gives no general proof.