The "spread" of Thompson's group $F$

Recall that a group $G$ is said to be $\frac{3}{2}$-generated if every non-trivial element $g\in G$ has a co-generator in $G$ (i.e., an element which together with $g$ generates $G$). Thompson's group $V$ was proved to be $\frac{3}{2}$-generated by Donoven and Harper in 2019. It was the first example of an infinite finitely presented non-cyclic $\frac{3}{2}$-generated group. In 2022, Bleak, Harper and Skipper proved that Thompson's group $T$ is also $\frac{3}{2}$-generated. Since the abelianization of Thompson's group $F$ is $\mathbb{Z}$, it cannot be $\frac{3}{2}$-generated. However, we recently proved that Thompson's group $F$ is "almost" $\frac{3}{2}$-generated in the sense that every element of $F$ whose image in the abelianization forms part of a generating pair of $\mathbb{Z}^2$ is part of a generating pair of $F$. A natural generalization of $\frac{3}{2}$-generation is the notion of spread. Recall that the spread of a group $G$ is the supremum over all integers $k$ such that every $k$ non-trivial elements of $G$ have a common co-generator in $G$. The uniform spread of a group $G$ is the supremum over all integers $k$ for which there exists a conjugacy class $C\subseteq G$ such that every $k$ non-trivial elements of $G$ have a common co-generator which belongs to $C$. In this paper we study modified versions of these notions for Thompson's group $F$.