Neumann's arc-count conjecture for non-Desarguesian projective planes

Let $\Pi$ be a finite projective plane of order $q$, and let $C_7(\Pi)$ denote its number of $7$-arcs. Write $C_7(q)$ for the corresponding value in the Desarguesian plane of order $q$.

Neumann's arc-count conjecture. The following equivalent inequalities should hold: for every finite non-Desarguesian plane,

$$C_7(\Pi) < (q^2 + q + 1)(q^2 + q)q^2 (q-1)^2 (q - 3)(q - 5)(q^4 - 20q^3 + 148q^2 - 468q + 498),$$

and, for every fixed odd prime power $q$ and every non-Desarguesian projective plane $\Pi$ of order $q$,

$$C_7(\Pi) < C_7(q).$$

These are reformulations of the conjecture that every non-Desarguesian finite projective plane contains a Fano subplane. The equivalence uses the source's theorem that $C_7(\Pi)+A_7(\Pi)$ depends only on $q$, together with the fact that Desarguesian planes of odd order contain no Fano subplanes. The source gives no resolution status.