The optimization process and recommendations:

1. Create Optimal Indexes (modified query below): The recommended indexes are an integral part of this optimization effort and should be created before testing the execution duration of the optimized query.

Optimal indexes for this query:

ALTER TABLE `MVAssignment` ADD INDEX `mvassignment_idx_associated_assignment_isdeleted` (`AssociatedObjectClass`,`AssignmentSubType`,`IsDeleted`);
ALTER TABLE `MVList_Stores` ADD INDEX `mvlist_stores_idx_mvlistid` (`MVListID`);
ALTER TABLE `MVMasterEntity_AttachedStores` ADD INDEX `mvmasterentity_att_idx_mvstoreid` (`MVStoreID`);
ALTER TABLE `TTT97a84e6f7eaa4bf3b2fbb8491517324c` ADD INDEX `ttt97a84e6f7eaa4bf_idx_id` (`ID`);

The optimized query:

SELECT
        mvas.MVStoreID AS StoreID 
    FROM
        MVAssignment AS ass_ 
    JOIN
        MVList_Stores AS list_stores 
            ON ass_.AssignmentValue = list_stores.MVListID 
    JOIN
        MVMasterEntity_AttachedStores mvas 
            ON mvas.MVStoreID = list_stores.MVStoreID 
    WHERE
        EXISTS (
            SELECT
                existsTT.ID 
            FROM
                TTT97a84e6f7eaa4bf3b2fbb8491517324c existsTT 
            WHERE
                existsTT.ID = ass_.AssociatedObjectID
        ) 
        AND ass_.AssociatedObjectClass = 'MVRep' 
        AND ass_.AssignmentSubType = 'ListID' 
        AND ass_.IsDeleted = 0