The forbidden-subdiagram characterization conjecture for k-terminal-minimality

A chord diagram is $k$-terminal-minimal if it is $k$-terminal and minimal with respect to that property under the reduction considered in the paper. The forbidden-subdiagram characterization conjecture. There is a nice infinite forbidden-subdiagram characterization of $k$-terminal-minimality. The source presents this as a preliminary expectation and gives no precise forbidden family or proof, so the conjecture remains open.