Degenerate diamonds require at least four leading terms

Let $q\geq 3$, let $\Pi$ be a projective plane of order $q$, and consider a $4\times4$ determinant initial form of its combinatorial incidence matrix. A degenerate diamond is a configuration $(X,Y;m,n)$ with

$$X,Y\notin m,n,\qquad m\cap n\in\overline{XY}.$$

Suppose the minimizer set of the determinant contains the swap pair associated with this degenerate diamond. Degenerate-diamond leading-term conjecture. For every projective plane of order $q\geq 3$, and in particular for every Desarguesian plane $\mathrm{PG}(2,q)$, such a degenerate diamond cannot occur when the determinant has exactly three valuation-minimizing permutations. Equivalently, carrying a degenerate diamond remainder should require at least four valuation-minimal determinant terms. The claim is motivated by the computational exclusion of the corresponding configuration in $\mathrm{PG}(2,3)$; its validity for general projective planes and larger orders remains open.