The Green-Tao theorem states that for every $n$, there is an arithmetic sequence of length $n$ consisting of primes.

For primes, $p$, let $P(p)$ be the maximum length of an arithmetic progression of primes whose least element is $p$.

Is it known whether $P(p)=p$ for every prime?

(This clearly generalizes the Green-Tao theorem, asserting that long progressions show up "as soon as possible." Note that $P(p) \leq p$ by viewing the progression mod $p$.)