Incomplete Tasks STAY on Timesheet

These are the only options for timesheets unfortunately. 

You will likely want to investigate the scenario of the task itself to troubleshoot the projected date being in the past to see if it can help solve for something. If task is not marked complete then the projected completion date updates based on certain criteria. 

• When the assignee updates the Commit Date:

  Projected Completion Date = Commit Date

• When the task does not have an updated Commit Date and the task has a forced constraint (Must Finish On) for the Planned Completion Date that is in the future:

  Projected Completion Date = Constraint Date

• When a task does not have an updated Commit Date, a forced constraint date in the future, or it has a Constraint Date in the past:

  Projected Completion Date = system calculation for the Completion Date based on the current progress and the work left to be done

https://experienceleague.adobe.com/en/docs/workfront/using/manage-work/tasks/task-information/differentiate-projected-estimated-dates#understand-when-projected-dates-update-on-tasks

Also something to consider is adjusting your timeframe for populating the data. Work that is within X of the timesheet's work range defines the number of weeks before and after the date range of the timesheet that contains dates of tasks and issues assigned to the user. You can extend this range to 4 weeks.

Final consideration: The user can "pin" the item to their timesheet and it will stay until they unpin it. 

• To manually pin an item that displays on the timesheet automatically, hover over the name of the item, then click the pin icon