“I don't see that happening any time soon.”
Your friend is writing a message on her mobile phone. It takes a long time for her to write it and there are a lot of spelling mistakes. She says that she wants a mobile phone that she can just speak into and it will type the message perfectly. You don't think that's possible, so you say this.
I don't see that happening any time soon.
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This means "I don't think it's going to happen." You use this when someone suggests that something will happen, but you don't believe it.
I don't think that's likely to happen.
"I don't see ___" can be completed with specific actions as well, instead of "that happening". For example:
I don't see Joe agreeing to do that.
This sentence means "I don't think that Joe will agree to do that."
The basic meaning of "not ___ any time soon" is "not soon". But it's usually used as an understatement. You say this when you don't think something will happen for a long time, or maybe ever.
This political turmoil isn't going to end any time soon.
I'm not going to win any awards for my cooking any time soon, I'm pretty sure.