Is it "inappropriate" to write in a manuscript "this problem is unsolved, though arguably progress has been made" and then reference your own work? Is it considered inappropriate to write in a manuscript "this problem remains unsolved, though arguably progress has been made" and then reference your own work?

Because you would be saying that your own work is the only progress being made on a subject... Is not it too arrogant?

The phrase is already written with another bibliographical reference. My question is just about whether I should replace and older reference with my own new work, which I of course think advances towards a solution.
I don't think I'd call it inappropriate, exactly, but I don't think it is the best way to word it.

Your use of the word "arguably" signals that the phrase "progress has been made" is subjective - and in particular, it's a subjective opinion coming from a source that is clearly not impartial (you).  Also, "progress has been made" is vague and not very helpful to the reader by itself - you're forcing them to go and look up your other paper if they want to know what kind of progress you mean.

So why not solve both problems, and replace this unhelpful subjective statement by a helpful objective one, that briefly describes what you actually did:


  This problem is unsolved.  However, in previous work [14], we showed that in the special case where the graph is symmetric, the conjectured statement is true to within a factor of 20.


Of course, if there is relevant work by other researchers, you should cite their work here as well, with a similar brief discussion.