Writing a Trigger when Lead is Closed

Hello, I am writing a trigger with the following requirements:

Write a trigger that automatically sets Tasks to the “Completed” status whenever their associated Leads are set to any Closed status. Make sure your trigger is bulkified - there should be no SOQL queries inside any loops.

I have written the following trigger - it works well if the task is 'Closed - Not Converted' however when I convert the lead, the task is not closing. Can you provide some guidance please?

trigger SetTaskCompletedOnLeadClosed on Lead (after update) {
//Create a set of ideas for no duplicates
    Set<Id> leadListIds = new Set<Id>();
    for(Lead l : Trigger.new){
    // Access the "old" record by its ID in Trigger.oldMap     
        if(l.status == 'Closed - Converted' || l.status == 'Closed - Not Converted'){
            leadListIds.add(l.id);
            system.debug('LeadListIds = '+leadListIds);
            }
        }
    List<Task> taskList = [SELECT id, subject, status, whoid FROM Task WHERE whoid in :leadListIds];
    system.debug('taskList = '+taskList);
    List<Task> taskToUpdate = new List<Task>();
    for(Task t : taskList){
        if(t.status != 'Completed'){
            t.status = 'Completed';
        }
         taskToUpdate.add(t);
         system.debug('taskToUpdate = '+taskToUpdate);
    }
    if(taskToUpdate.size()>0){
    update taskToUpdate;
    }
}