The extremal alternating-sign conjecture for transitive tournaments

Let $TT_n$ be a transitive tournament on $n$ vertices. For $\alpha=(\alpha_1,\alpha_2,\dots,\alpha_s)\in\mathbb{K}_s$ satisfying

$$\sum_{i=1}^s\alpha_i=n-1,$$

Extremal alternating-sign conjecture. If $n$ is even, then

$$f_{TT_n}(1,-1,1,\dots,-1,1)\geq f_{TT_n}(\alpha_1,\alpha_2,\dots,\alpha_s),$$

where $(1,-1,1,\dots,-1,1)$ has $n-1$ components. If $n$ is odd, then

$$f_{TT_n}(1,-2,1,-1,1,\dots,-1,1)\geq f_{TT_n}(\alpha_1,\alpha_2,\dots,\alpha_s),$$

where $(1,-2,1,\dots,1)$ has $n-2$ components. The conjecture proposes that these alternating-sign tuples maximize the number of Hamiltonian paths counted by $f_{TT_n}$ among all admissible tuples. The preceding computations verify the analogous extremal behavior for the tested values of $n$, while the general assertion remains open.