Quadrisection bound for convex odd-gons

Let $R$ be a convex polygon with $2n+1$ sides. A quadrisection is a partition of $R$ into four regions of equal area by three concurrent cevians. Quadrisection bound conjecture. The polygon $R$ has at most $2n+1$ quadrisections. Further, if $R$ is sufficiently close to the regular $2n+1$-gon, then $R$ has $2n+1$ quadrisections.

This conjecture extends the triangle quadrisection problem to convex polygons and predicts both a sharp upper bound for odd-gons and its attainment near the regular polygon. The supplied text gives no resolution, so the conjecture remains open.