On top of my own company's, I'm constantly signed in to 21 Slack workspaces mostly belonging to clients, and the endless notifications are starting to become a real problem, particularly when people use @channel or @here all the time. Add into that the Slack's notification sounds are some of the most grating noises to hit my ears since Skype's. If I'm concentrating on a complex task and Slack starts going off I come very close to throwing my laptop out the window.
That can work for some slacks and some channels. But with others, not so much. When you're being paid to pay attention to client needs, there's only so much you can get away with.
But Slack still creates the expectation of immediacy in a way that other tools don't. Handling the same number of clients via email is a whole different experience. And Slack makes it very hard for people to filter out noise.
To my mind, Slack is to virtual work what open offices are to physical collaboration. They can be used well, but the default is a continuous drain on attention.
In short, I hope you succeed in solving this.