How to swap the paths of two records (and their children) using a postgresql ltree?

Question

How do I swap two record's paths/positions in an ltree using postgres, where all children follow? There is a uniqueness constraint on the path.

Given the following records on a pages table:

id: 1,  path: "1",
id: 2,  path: "1.1",
id: 3,  path: "2",
id: 4,  path: "2.1",
id: 5,  path: "2.2",
id: 6,  path: "2.2.1",
id: 7,  path: "2.2.1.1",
id: 8,  path: "2.3",
id: 9,  path: "2.3.1",
id: 10, path: "2.3.1.1",
id: 11, path: "2.3.1.2",
id: 12, path: "2.3.1.3",
id: 13, path: "2.3.1.4",
id: 14, path: "2.4",
id: 15, path: "3",
id: 16, path: "3.1",
id: 17, path: "3.2",
id: 18, path: "3.2.1",
id: 19, path: "3.2.2",
id: 20, path: "3.2.3",

I can move record 15 (and children 16-20) to anywhere in the tree using the following:

--  $1 is newpath, $2 is oldpath
UPDATE "pages" SET "path" = $1  WHERE "pages"."id" = 15;
UPDATE "pages" SET path = $1 || subpath(path, nlevel($2)) WHERE (pages.path <@ $2) AND (pages.path != $1)

But what is the best way to to swap the positions of two records (and children)? For example:

• changing the paths of ids 5 (5 keeps children 6-7) and 8 (8 keeps children 9-13)
• changing the paths of ids 3 (3 keeps children 4-14) and 15 (15 keeps children 16-20)

I saw this post, where the author swaps a value of two rows with a temporary value (in their case they negate). Should I do a similar thing, but using the query from before three times? Is there a better way which doesn't require 6 UPDATE calls?