It's encouraging to see the response from EE at the end of the post.

Edit: Wait. That tweet is from @MyEECare (https://twitter.com/MyEECare) not @EE, and the former has the same bio as the latter, except for the "Entirely fictional" at the end, so it looks like that's a spoof account, not an official EE one.

EE have a history of poor social media management as they have tweeted DMs publicly before. The last time they said they were putting process in place to stop it happening but it only needs one password. Besides which if somebody hacked the account the passwords could be harvested very easily!