One-step separation conjecture for genus Tutte polynomials

Let $g$ be a positive integer, and let $\{M_i\}$ and $\{N_i\}$ be sequences of matroids. One-step separation conjecture. There exist two infinite sequences of matroids $\{M_i\}$ and $\{N_i\}$ such that

$$T^{(g)}(M_i)=T^{(g)}(N_i)$$

and

$$T^{(g+1)}(M_i)\neq T^{(g+1)}(N_i).$$

This conjecture asks whether the constant controlling the genus needed to distinguish infinite families can be reduced to one. The surrounding proposition establishes separation at a genus of the form $\lceil c_1g+c_2\rceil$; the conjecture that separation occurs already at $g+1$ remains open.