Rational minimum-rank conjecture for sign patterns with at most three zeros per column

Let ${\cal A}$ be a sign pattern, and let $mr({\cal A})$ denote its minimum rank over the reals while $mr_{\mathbb Q}({\cal A})$ denotes its minimum rank over the rationals. Rational minimum-rank conjecture. If

$$m r({\cal A})=3,$$

and each column of ${\cal A}$ contains at most three zero entries, then

$$mr_{\mathbb Q}({\cal A})=mr({\cal A}).$$

The preceding theorem establishes the analogous conclusion when each column contains at most two zeros; the conjecture addresses the remaining three-zero case, motivated by the absence of known non-rationally realizable point-line configurations with at most three points on each line.