There are even stronger conjectures in the literature:
- D.R. Heath-Brown, Almost-primes in arithmetic progressions and short intervals. Math. Proc. Cambridge Philos. Soc. 83 (1978), no. 3, 357–375. http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2079092
quoting from the first few lines: "... if $(l,k)=1$, there exists a prime $p$ satisfying $$p \equiv l \bmod k, \quad p\leq k^2, \quad (k\geq 2).$$ Indeed the bound for $p$ may presumably be reduced to $p \ll k(\log k)^2$. "
Actually, Heath-Brown unconditionally proves that there are many (i.e. the expected number) $P_2$ numbers smaller than $k^{2-\delta}$, for small $0\leq \delta \leq 0.035$. Maybe this gives a partial result for your injectivity question.
Further references:
A paper by A. Granville (Least primes in arithmetic progressions. Th'{e}orie des nombres (Quebec, 1987), 306–321, de Gruyter, Berlin, 1989) http://www.dms.umontreal.ca/~andrew/1989.html
points to an even more explicit version of a least prime conjecture: following a heuristic of Wagstaff, see http://www.ams.org/journals/mcom/1979-33-147/S0025-5718-1979-0528061-7/ , McCurley observed that this heuristic implies: $\lim\sup_{k\rightarrow \infty} \frac{p(k)}{k(\log k)^2} =2$.
A final comment: Your function $f(p)$ is (presumably) injective on the primes, but not on all integers. (Say 11 is the least prime 1 mod 10, and the least 1 mod 5). This may explain why others, possibly, did not study the question from your point of view.