The forbidden-pattern characterization conjecture for one-stack one-queue layouts

Let an ordered matching have a fixed non-separated layout. A $1$-stack $1$-queue layout uses one stack and one queue, and a mixed layout on $1$ page uses one page whose edges may be assigned to stacks or queues. The twelve and eight patterns referred to below are the patterns depicted in the source's figures.

Forbidden-pattern characterization conjecture. The ordered matching admits

$$\text{a $1$-stack $1$-queue layout if and only if it avoids the 12 patterns depicted in the source's figure,}$$

and admits a mixed layout on one page if and only if it avoids the 8 patterns depicted in the source's figure.

This conjecture proposes complete finite obstruction sets for the two smallest non-separated layout classes. It is consistent with the preceding computational enumeration of twelve and eight critical matchings, but the supplied source gives no proof or resolution.