1) Create template TEM1 and create there triggers TR1, TR2 and TR3
2) create dependency so that TR1 depends on TR2
3) create autoregistration that adds the host and links template TEM1
4) start server and agent
5) Result - new host H1 was created, template TEM1 was linked and trigger TR1 on host correctly depends on TR2
6) stop the server and the agent
7) Unlink (not unlink and clear) the template TEM1 from the the host H1
8) Update template TEM1 so that TR1 now depends on TR3
9) Update agent host metadata and restart server and agent
10) autoregistration occurs, template TEM1 again gets linked to host H1, but the trigger TR1 on host still depends on TR2, not on TR3 (like a trigger on template)

Also, the same situation would occur if a trigger dependency would be deleted on a template - the change would not get propagated.