Carousel homomorphism conjecture for odd carousel tournaments

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

Carousel homomorphism conjecture. For every $n\in\mathbb{N}$, the carousel homomorphism maximizes the density of $R_{2n+1}$:

$$\max\{\phi(R_{2n+1}):\phi\in\operatorname{Hom}^+({\mathcal{A}}^0,\mathbb{R})\}=\phi_R(R_{2n+1}).$$

The conjecture is motivated by the structural similarity between odd carousel tournaments and $R_4$; its validity for all $n$ remains open.