The matched-diagram conjecture for links

A matched diagram is a link diagram obtained from a signed, two-color planar graph by the construction described in the source. An oriented link is $t_3$-move equivalent to another link if it can be transformed into it by $t_3$ moves.

Matched-diagram conjecture. (i) Not every link has a matched diagram. (ii) Every oriented link is $t_3$-move equivalent to a link with a matched diagram.

The first assertion is presented as a conjectural obstruction to the existence of matched diagrams for all links, while the second proposes that matched diagrams nevertheless represent every oriented-link type up to $t_3$ moves. The source does not give a resolution of either assertion.