The linked chart shows that there were none in the 20-24 age range during the during the recent few years. Is the entire population vaccinated? If not (the article doesn't claim this), then the fact that no one in that age range died (and only 5 in the entire under-30 cohort) tends to indicate that it was not a very high base rate.
Are there other sources that show data going back to the 1970s? Probably! I didn't go searching for them. I looked at what was linked above and saw there were very few. As I said, the Guardian journalist didn't include a base rate, which surely would have been included if it bolstered the argument.
EDIT: I just scrolled down further and saw that even the chart that shows trends over time (which I hadn't seen before, having stopped scrolling earlier) doesn't support your point. It shows there were roughly .2 deaths per year per 100k. Not having any deaths in 20-24 for 3 years is not a statistically significant difference, I would imagine, than the .2 figure. Also, there are undoubtedly other cancer-related advances that have made it less likely that a young woman would die of any kind of cancer.
And the data regarding under-30 deaths is muddled because the next bucket up is 25-34, and we don't know what it is up to 29.
Lastly, at the bottom there's this disclaimer, which makes it even harder to tell what's going on with small numbers:
> Note: Non-zero counts of 5 or less are suppressed and presented as 5.
If you have another source, please feel free to share. What we've seen so far (nothing in TFA, nothing of import in the commenter's linked data) isn't remotely compelling.