Final answer:
The Byzantine Empire was slowly losing its hold over its territories, and the notion of Germanic tribes settling within its former territories is most supported by the historical context. Claims about Lombards invading through the Black Sea or the Visigoths slowly invading the Byzantine Empire are not as strongly substantiated.
Step-by-step explanation:
Based on the provided historical context, it is most likely true that the Byzantine Empire was slowly losing its hold over its territories. The sources highlight a period of contraction and struggle for the Byzantine Empire, with losses in the east to Islamic conquerors and challenges to their authority from various enemies, including Germanic tribes and later Turkic tribes. In addition, another true statement is that different Germanic tribes wanted to be part of the Byzantine Empire, at least in the sense that they sought to establish kingdoms within territories that had previously been under Roman control, as indicated by the Goths settling in Gaul and Spain, and later the Lombards in Italy.
While the Lombards did indeed invade parts of the Empire, the statement about the Lombards invading through the Black Sea is not supported by the context provided. Finally, while the Visigoths did engage in conflicts with the Byzantine Empire, the statement about the Visigoths slowly invading the Byzantine Empire is not the most accurate depiction of the historical events, as they were more focused on establishing their own rule in territories like Gaul and Spain.