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I know I've heard/read people discuss this before but can't find anything on it now...
How does search suggestions work in Google? Does anyone have any experience with trying to influence the suggestions?
I've got one such situation and what I'm trying to do for the client is rank well on the suggested query - at least getting the client's point of view and messaging seen. I've got three results on the bottom of page 1 now, and of course I'm working to improve on that.
I had a week of hope in early October when the identical "related searches" entry vanished from the bottom of the SERP - but it came back, and the Google Suggestion never went away.
It got me worried...
Luckily so far the related searches at the bottom are all neutral.