The maximum commutative subsemigroup conjecture for full transformation semigroups

Let $X$ be a set, and let $\mathcal{T}(X)$ denote the full transformation semigroup on $X$. For the notation $N^{X}_{x_1,\ldots,x_t}$ and the functions $\xi$ and $\alpha$, use the definitions introduced in the paper. Maximum commutative subsemigroup conjecture. Suppose that $|X|\geqslant 7$. Then the maximum size of a commutative subsemigroup of $\mathcal{T}(X)$ is $(|X|)\xi+1$. Moreover, the maximum-order commutative subsemigroups of $\mathcal{T}(X)$ are precisely the semigroups

$$N^{X}_{x_1,\ldots,x_t}\cup\{\operatorname{id}_{X}\},$$

where $t=(|X|)\alpha$ and $x_1,\ldots,x_t\in X$ are pairwise distinct. This conjecture would resolve the open problem of determining and characterizing the largest commutative subsemigroups of the full transformation semigroup, and would consequently provide information about the corresponding clique-number problems.