Can there exist three integers p, q and r with 2 ≤ p < q < r, that satisfy each of the following conditions?

(i) p^{2} -1 is divisible by each of q and r, and:

(ii) r^{2} -1 is divisible by each of p and q.

By the way, note that my proof (see first posting) is stronger than the problem requested.

I have proven that there do not exist three integers p, q and r

with 2 ≤ p and p < q and p < r, that satisfy each of the following conditions:

(i) p

^{2} -1 is divisible by each of q and r, and:

(ii) r^{2} -1 is divisible by q.