Uniqueness conjecture for the carousel homomorphism maximizer

Let $R_{2n+1}$ be the odd carousel tournament, let $\phi_R$ be the carousel homomorphism, and let $\operatorname{Hom}^+({\mathcal{A}}^0,\mathbb{R})$ denote the positive homomorphisms of the tournament algebra.

Carousel maximizer uniqueness conjecture. For every $n\geq2$, a homomorphism $\phi\in\operatorname{Hom}^+({\mathcal{A}}^0,\mathbb{R})$ maximizes the density of $R_{2n+1}$ if and only if $\phi=\phi_R$. This is presented as a consequence of the preceding maximum-density conjecture and asserts uniqueness of the maximizer for all $n\geq2$; it remains open.