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This used to be a straightforward process and was well explained both Here and Here. The short of it is that there used be a 'Join Our Slack' link on right slide-in menu and now it is no longer. The Herokuapp is also down.
So how, oh how my dear friends do we get invited to Steem's Slack? :)
I will update this post with all working solutions.
Please let me know in the comments who else you think I should page to increase the chance we get a solution!