Realization of all circular extensions by cursor-jump orders

Prove that, for every connected downward closed graph G, every circular extension of the associated partial cyclic order Z′_G occurs as the total cyclic order of cursor jumps for some parameter choice (a₁,…,a_N,p₁,…,p_N) in P_G.

Background

For parameters whose downward closed graph is connected, the stationary evolution induces a total cyclic order describing the order in which cursors jump. The paper constructs a map from total cyclic orders to connected downward closed graphs and defines a partial cyclic order Z′_G whose circular extensions are exactly the fiber over G.

The conjecture asserts surjectivity at the level of parameter realizations: every combinatorially admissible circular extension should be realized dynamically by some parameters in the corresponding region. The authors report numerical verification for N≤4.