The conjecture that the even and odd vector-space limits differ

Let $p$ be an odd prime and let $g$ be a natural number. Define

$$L_e=\lim_{k\to\infty}\frac{\eta_g\left(\mathbb{F}_p^{2k}\right)}{\sqrt{p^{2k}}},\qquad L_o=\lim_{k\to\infty}\frac{\eta_g\left(\mathbb{F}_p^{2k+1}\right)}{\sqrt{p^{2k+1}}}.$$

Theorem 1 establishes that these limits exist and are positive real numbers. Even–odd limit conjecture. The two limits are distinct:

$$L_e\ne L_o.$$

The conjecture concerns the dependence of the normalized cardinality of $g$-difference sets in finite vector spaces over $\mathbb{F}_p$ on the parity of the dimension. The preceding theorem proves existence and positivity of the two limits, but their inequality remains open in the source.