The semiprime $87 = 3*29$ has a curious property: it's the fact that both

$87^2 + 29^2 + 3^2 = 8419$

and

$87^2 - 29^2 - 3^2 = 6719$

are prime numbers.

This intrigued me and led me to wonder if there are other semiprimes with the same property, and I found that

$21 = 3*7$ is another example, since both

$21^2 + 7^2 + 3^2= 499$

and

$21^2 - 7^2 - 3^2 = 383$ are prime numbers

So the following question arises: Are there infinitely many prime numbers $p$ and $q$, with $p \neq q$, such that both

$(pq)^2 + p^2 + q^2$

$(pq)^2 - p^2 - q^2$

are also primes?

Does this follows from some known theorem or conjecture?