Converse parametrization conjecture for Brunovsky transformations

Let $(A_s,B_s)$ be a linear system in staircase form, and let $(T,F,G)$ be a Brunovsky transformation that maps it to Brunovsky form. Let $C$ be an observation matrix satisfying the rank constraints in equation~; such a matrix defines a parametric triplet $(T,F,G)$ through the formulas in equation~.

Converse parametrization conjecture. If a Brunovsky transformation $(T,F,G)$ transforms the staircase linear system into Brunovsky form, then there exists an observation matrix $C$ satisfying the stated rank constraints such that $C$ defines $(T,F,G)$ by the parametric transformation formulas.

This is the converse of the preceding parametrization theorem and would establish that the proposed observation-matrix construction captures every Brunovsky transformation. The supplied text gives no resolution, so the conjecture remains open.