A thought: there's no way the project is staying hidden forever. Remember, if we succeed then this is going to be a major infrastructure project all across the Old World. The "let's build a magic tower" chapters showed that this sort of thing requires lots of weird imports and an actual workforce, not just two wizards sitting in a ditch somewhere. And the "let's examine a waystone" chapter showed us that even being able to semi-safely work in the right spot will often require a military escort, which isn't exactly inconspicuous either.
And if we get that far, then this will definitely be noticed by surrounding institutions, be a big deal, etc. Maybe it won't be perceived as "the biggest thing ever" (investment in boring, foundational infrastructure never is unless there's a "5G towards are mind-control"-type scare), but people will know and will poke at it.
In such a situation, absolute secrecy is a pipe dream. Instead, what we want is plausible deniability and political weight. That's the point of being "less obvious": people won't notice offhand, and when it is noticed raising a stink will be a stupid decision for the institution in question.