TEB planner tuning help to follow a path strictly

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So what I am trying to do, is let the robot follow a global path very strictly.
I know this path doesn't make sense, as it's not taking the shortest route, etc, but let's just say it's like a street the robot has to follow.

My question is, if it is even possible to limit the planner in such a way, that it follows the path and doesn't take a shortcut.
I tried to enable via points and decreased the global plan lookahead but it doesn't affect planning at all.

Also when looking at the second gif, the robot moves back to the beginning of the path.
But for this case, I thought the planner is pruning the already passed global path with calling this function pruneGlobalPlan() , similar to dwb is doing it.

controller_frequency: 10.
controller_plugin_ids: ["FollowPath"]
controller_plugin_types: ["teb_local_planner::TebLocalPlannerROS"]
min_theta_velocity_threshold: 0.0001
min_x_velocity_threshold: 0.0001
min_y_velocity_threshold: 0.0001
FollowPath.acc_lim_theta: 3.
FollowPath.acc_lim_x: 2.
FollowPath.acc_lim_y: 0.
FollowPath.allow_init_with_backwards_motion: False
FollowPath.cmd_angle_instead_rotvel: False
FollowPath.complete_global_plan: True # true prevents the robot from ending the path early when it cross the end goal
FollowPath.costmap_converter_plugin: "costmap_converter::CostmapToPolygonsDBSMCCH"
FollowPath.costmap_converter_spin_thread: True
FollowPath.costmap_obstacles_behind_robot_dist: 1.0
FollowPath.delete_detours_backwards: True
FollowPath.detours_orientation_tolerance: 1.5708
FollowPath.dt_hysteresis: 0.05
FollowPath.dt_ref: 0.3
FollowPath.dynamic_obstacle_inflation_dist: 0.6
FollowPath.enable_homotopy_class_planning: True
FollowPath.enable_multithreading: True
FollowPath.exact_arc_length: False
FollowPath.feasibility_check_no_poses: 5
FollowPath.footprint_model.radius: 0.6
FollowPath.footprint_model.type: "circular"
FollowPath.force_reinit_new_goal_dist: 1.
FollowPath.free_goal_vel: False
FollowPath.global_plan_overwrite_orientation: True
FollowPath.global_plan_prune_distance: 2.0
FollowPath.h_signature_prescaler: 1.
FollowPath.h_signature_threshold: 0.1
FollowPath.include_costmap_obstacles: True
FollowPath.include_dynamic_obstacles: True
FollowPath.inflation_dist: 0.1
FollowPath.is_footprint_dynamic: False
FollowPath.legacy_obstacle_association: False
FollowPath.length_start_orientation_vector: 0.4
FollowPath.map_frame: "map"
FollowPath.max_global_plan_lookahead_dist: 0.5
FollowPath.max_number_classes: 5
FollowPath.max_ratio_detours_duration_best_duration: 3.
FollowPath.max_samples: 20
FollowPath.max_vel_theta: 1.
FollowPath.max_vel_x: 1.3
FollowPath.max_vel_x_backwards: 0.3
FollowPath.max_vel_y: 0.0
FollowPath.min_obstacle_dist: 0.1
FollowPath.min_resolution_collision_check_angular: 3.14159
FollowPath.min_samples: 3
FollowPath.min_turning_radius: 0.0
FollowPath.no_inner_iterations: 5
FollowPath.no_outer_iterations: 4
FollowPath.obstacle_association_cutoff_factor: 5.
FollowPath.obstacle_association_force_inclusion_factor: 1.5
FollowPath.obstacle_cost_exponent: 1.
FollowPath.obstacle_heading_threshold: 0.45
FollowPath.obstacle_keypoint_offset: 0.1
FollowPath.obstacle_poses_affected: 25
FollowPath.odom_topic: "odom"
FollowPath.optimization_activate: True
FollowPath.optimization_verbose: True
FollowPath.oscillation_filter_duration: 10.
FollowPath.oscillation_omega_eps: 0.1
FollowPath.oscillation_recovery: True
FollowPath.oscillation_recovery_min_duration: 10.
FollowPath.oscillation_v_eps: 0.1
FollowPath.penalty_epsilon: 0.1
FollowPath.publish_feedback: False
FollowPath.roadmap_graph_area_length_scale: 1.
FollowPath.roadmap_graph_area_width: 2.
FollowPath.roadmap_graph_samples: 15
FollowPath.selection_alternative_time_cost: False
FollowPath.selection_cost_hysteresis: 1.
FollowPath.selection_obst_cost_scale: 100.
FollowPath.selection_prefer_initial_plan: 0.95
FollowPath.selection_viapoint_cost_scale: 1.
FollowPath.shrink_horizon_backup: True
FollowPath.shrink_horizon_min_duration: 10.
FollowPath.simple_exploration: False
FollowPath.switching_blocking_period: 0.
FollowPath.teb_autosize: 1.
FollowPath.global_plan_viapoint_sep: 1. # If positive, via-points are extracted from the global plan (path-following mode). The value determines the resolution of the reference path (min. separation between each two consecutive via-points along the global plan, if negative: disabled). Refer to parameter weight_viapoint for adjusting the intensity.
FollowPath.via_points_ordered: True # If true, the planner adheres to the order of via-points in the storage container
FollowPath.viapoints_all_candidates: True
FollowPath.visualize_hc_graph: True
FollowPath.visualize_with_time_as_z_axis_scale: 0.
FollowPath.weight_acc_lim_theta: 1.
FollowPath ...