Distinct even and odd asymptotic constants over finite fields

Prove or disprove that the limiting constants \(L_e\) and \(L_o\) for normalized \(g\)-difference bases in even- and odd-dimensional vector spaces over \(\mathbb{F}_p\) are unequal, namely that \(L_e\ne L_o\).

Background

For a fixed odd prime $p$ and natural number $g$, the paper proves that the normalized minimum sizes of $g$-difference bases in $\mathbb{F}_p^{2k}$ and $\mathbb{F}_p^{2k+1}$ converge separately to positive limits $L_e$ and $L_o$. Unlike the integer case, the authors cannot establish whether the parity-dependent limits coincide and explicitly formulate their expected inequality as a conjecture.