Primes Between Squares of Primes Is this problem still open? I know that Henri Brocard conjectured that there are at least four primes in the interval between each pair of consecutive squares of primes from nine onward. < I know that Brocard's conjecture remains open. I am asking if there is yet a proof for only one prime between each pair of consecutive squares of primes. Is this a problem with or without a name? Where can I find a reference about it or any information on progress made on it?

A reference, among others, on such problems is the paper On Legendre’s, Brocard’s, Andrica’s, and Oppermann’s Conjectures by German Andres Paz. His "Conjecture $1$" related to Brocard's conjecture as follows. Let denote $p_n$ and $p_{n+1}$ two successive primes greater than $2$. Since $p_{n+1}-p_n\ge 2$, we know that there is a positive integer $k$ with $p_n
So I think, Conjecture $1$, Brocard's conjecture, and "your conjecture" (which is a variant of these) are still open.