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Waves are comments.

They are in a sense, I agree, but if you take waves as short content platform then it'll be better to have such feature. For example inleo #threads appear in profile page under Threads section and it is easy to review thread written. Same should be here.

I thing the normal notification does a great job also

At we have a waves section in profile page.

I don't think it's necessary though since we get it as a new comment and we can continue the conversation there

If you quickly need to look at all waves threads, it is a good UX in terms of website UI.

Am using the mobile app at the moment so I can't phantom how the web looks like

There are some features that you can only use on desktop version. But, Ecency points can only be used on App I guess.

Will try it out on web to see the new features, loving the mobile app version though

On website, we open waves replies into waves page. On mobile app, we don't have that functionality yet.

For some waves I see replied back in Waves, but mostly it opens in as a regular comment unless I click the parent post.

Oh now I see what you mean, we will check this bug.

That'll be great too.