The groups are actually submited when the "continue" button is clicked (if it is group by group or question by question). Of course the survey will not be marked as completed, but the answers should be stored in the database, even if the respondent doesn't proceed any further (as long as they click the continue button).
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Provide your LS version and where it is installed (own server, uni/employer, SaaS hosting, etc.).
Always provide a LSS file (not LSQ or LSG).
Note: I answer at this forum in my spare time, I'm not a LimeSurvey GmbH employee.