I tried twice. Am I closer to my luck?
If it works, yes.
I tested in the Organizer using three times about 20 years ago, and that worked. I have not tested since. I have no reason to think that this has changed, but it could have.
If you have styles that are based on other styles, as is common with heading styles, three times works.
You simply click on the copy button three times with the same styles selected. You can use Ctrl+Click to select multiple styles in the Organizer.
Another consideration is whether your numbering is linked to the styles. From your screenshot, it would appear that the first three levels are so linked.
See How to create numbered headings or outline numbering in Ribbon Versions of Word by Shauna Kelly.
If you simply pressed one of the numbering buttons to apply numbering to your styles, this does not necessarily link the numbering to the style.
I routinely copy styles linked to numbering using the Organizer and it works well.
A Global StyleSheet in Microsoft Word?
Outline Heading Styles Global Stylesheet