From Developmental Biology, 6th edition I learned the following:
In humans, the oogonia divide to form a limited number of egg precursor cells. In the human embryo, the thousand or so **oogonia** divide rapidly from the second to the seventh month of gestation to form roughly 7 million **germ cells**. After the seventh month of embryonic development the number of germ cells drops and most oogonia die during this period, while the remaining oogonia enter the first meiotic division. These latter cells, called the **primary oocytes** , progress through the first meiotic prophase until the diplotene stage, at which point they are maintained until puberty.
Hence, at puberty the cells from the lineage of oogonia are called primary oocytes. So from a terminological perspective, you are correct that **no oogonia are thought to be left at puberty**.