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That is one route, yes.
I tend to go a bit further with my synths though, before going to my nagura progression.
Usually I'd go to a 5 or 8K grit synthetic.
Not that I couldn't go directly to a...

The name Nakayama does add to the price, regardless of performance.
As to wether or not you would get a better edge from one, compared to the one you have, who knows?
Natural stones are weird like...

That is a good looking base for you stone!
Nice job, that should last you a lifetime:)