First-order nonvanishing obstruction for rank-three incidence lifts

Let $\Pi$ be a projective plane of order $q$, let $U$ be a rank-$3$ residue zero-pattern model satisfying the identity-pattern minor constraints, and let $T_U$ denote its tangent space. First-order nonvanishing obstruction. For sufficiently large $q$, no such model admits a tangent direction $V\in T_U$ satisfying

$$V_{p\ell}\neq 0\qquad\text{for every incidence }p\in\ell.$$

Equivalently, any putative rank-$\leq 3$ lift of the incidence matrix $I(\Pi)$ should force at least one incidence entry to acquire valuation at least $2$. This would strengthen the paper's established first-order obstructions by ruling out a tangent direction that is nonzero on every incidence position; the global claim remains open.