There are even stronger conjectures in the literature:

1) 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.htm

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.

Update:
Computing the first values of least primes 1 mod p, and checking the OEIS
http://oeis.org/A035095

where we find the comment:

"Formula: According to a long-standing conjecture (see the 1979 Wagstaff reference), a(n) <= prime(n)^2+1. This would be sufficient to imply that a(n) is the smallest prime such that greatest prime divisor of a(n)-1 is prime(n), the n-th prime: A006530(a(n)-1)=A000040(n). This in turn would be sufficient to imply that no value occurs twice in this sequence.
- Franklin T. Adams-Watters, Jun 18 2010"

It seems, the last half sentence is the injectivity you are looking for, based
on an explicit bound of your $f(p)$.

Further, in http://oeis.org/A066674
we find the discussion, if the two sequences above are the same, with related comments by Poonen and Bach.