The regular-vector reconstruction conjecture for biplanes

Let ${\cal B}$ be a biplane, let $a_1,a_2,\ldots,a_m\in{\cal \grave{M}}^v$ be vectors obeying the regular-vector property $|f_{a,j}^{-1}(2)|=q(a)$ for $j\in Q(a)$, and let

$${\cal \grave{M}}^{v({a_1}{a_2}\ldots{a_m})}:={\cal \grave{M}}^{v(\hat{a_1})}\cup\ldots\cup{\cal \grave{M}}^{v(\hat{a_m})}\cup{\cal \grave{M}}_i^v,\qquad i\in E.$$

Here ${\cal \grave{M}}^{v(\hat a)}$, ${\cal \grave{M}}^{v(a)}$, $E$, $Q(a)$, and the level sets $f_{a,j}^{-1}(2)$ are defined as in the preceding construction, and a $2$-space is the associated vector space used to represent biplanes. Regular-vector reconstruction conjecture. There exists a natural number $r>0$ such that the $2$-spaces ${\cal \grave{M}}^{v({a_1}{a_2}\ldots{a_m})}$ and ${\cal \grave{M}}^v$ contain biplanes of the same isomorphism class. The claim is presented as a conjecture supporting the paper's computational recognition method for biplanes. Because the source does not state what is known about this assertion, its resolution status is unclear.