Conjecture on the exact value of f(q+2)

Let $q$ be the order of the finite projective plane, and let $f(n)$ denote the minimum number of lines whose odd-intersection point set has size $n$. Conjecture.

$$f(q+2)=2q-2.$$

The paper has established the upper bound and proves the equality in the preceding theorem for the relevant small values of $q$; the equality is conjectured in general.