Attachment to one frozen conical exterior

Use the cone wedge of angle $3\pi /2$, and place the patch in its first quadrant by

The translated $K$ lies strictly inside that quadrant, away from the axis and seam. On the rest of the cone use $T=H+s$. By prop:cap, the pieces agree on an open boundary neighbourhood.

Here is a finite exterior cellulation with exactly the required patch boundary subdivision. Put

In the first quadrant cut by $X\in A$ and $Y\in B$. In the second quadrant, using coordinates $(-X,Y)$, use cuts $(-X)\in A$, $Y\in B$. In the third quadrant, using $(-X,-Y)$, use cuts $(-X)\in A$, $(-Y)\in A$. The positive $Y$-axis subdivisions match through $B$, the negative $X$-axis subdivisions match through $A$, and the two wedge-seam subdivisions both use $A$. Keep the cells outside the translated $K$ and replace its four grid rectangles by the eight patch cells. This gives finitely many cell labels; unbounded exterior cells are allowed.

Use shifted cone developments

Time translation commutes with the seam rotation, so the particle word is unchanged. From a collar chart to the first-quadrant cone development, the transition is the fixed spatial translation