The extremal $K_4$-free graph conjecture for fractional separation dimension

Let $G$ be an $n$-vertex graph that does not contain $K_4$, and let $\pi_f(G)$ denote its fractional separation dimension. For $n\ge 10$, consider the complete tripartite graph

$$K_{1,\lfloor (n-1)/2\rfloor,\lceil (n-1)/2\rceil}.$$

The extremal $K_4$-free graph conjecture. For $n\ge 10$, the $n$-vertex graph not containing $K_4$ that maximizes $\pi_f$ is $K_{1,\lfloor (n-1)/2\rfloor,\lceil (n-1)/2\rceil}$. The conjecture is motivated by computations verifying the extremum among tripartite graphs up to $14$ vertices, while the supplied text does not establish the claim for all $n\ge 10$.