Asymptotic sparsity conjecture for digraph placement game values

Let $F(b)$ denote the largest number of vertices needed to construct a game value born by day $b$, and let $g(b)$ be the comparison function defined in the paper. Asymptotic sparsity conjecture.

$$F(b)=o(g(b)).$$

The conjecture asserts that the vertex complexity $F(b)$ grows asymptotically more slowly than $g(b)$. The paper establishes $F(b)<g(b)$ for $b\leq4$ and notes that the observed gap expands rapidly, but a proof would require a new upper-bound method or a sufficiently strong estimate involving the auxiliary function $a(b)$.