TITLE: Analog of Cramer's conjecture for primes in a residue class
QUESTION [7 upvotes]: Let $q$ and $r$ be fixed coprime positive integers, 
$$
1 \le r < q, \qquad \gcd(q,r)=1.
$$
Suppose that two prime numbers $p$ and $p'$, with $p<p'$, satisfy
$$
p \equiv p' \equiv r \ ({\rm mod}\ q),  \tag{1}
$$
and no other primes between $p$ and $p'$ satisfy $(1)$.
Then we have the following 
Naive generalization of Cramer's conjecture to primes in residue class $r$ mod $q$:
$$
p'-p ~<~ \varphi(q)\,(\ln p')^2.   \tag{2}
$$
(PrimePuzzles Conjecture 77, A. Kourbatov, 2016). 
See arXiv:1610.03340, "On the distribution of maximal gaps between primes in residue classes" for further details, including the motivation for the $\varphi(q)$ constant.
Here, as usual, $\varphi(q)$ denotes Euler's totient function.
Note: In the inequality $(2)$ we take the logarithm of the prime $p'$ at the larger end of the "gap".
Very few counterexamples to $(2)$ are known; see Appendix 7.4 in arXiv:1610.03340. Definitely no counterexamples for $q=2, \ p<2^{64}$; also none for $1\le r < q \le 1000$, $ \ p<10^{10}$.
This conjecture (mostly in a less-naive "almost always" form) is mentioned in the following OEIS sequences listing maximal (record) gaps between primes of the form $p=qk+r$, $ \ \gcd(q,r)=1$:
A084162,
A268799,
A268925,
A268928,
A268984,
A269234,
A269238,
A269261,
A269420,
A269424,
A269513,
A269519.
Question 1: Find a counterexample to conjecture $(2)$.
Question 2: Find a counterexample to $(2)$, with prime $q$ and prime $r$. 
Question 3: Find a counterexample to $(2)$, with
$$
{p'-p \over \varphi(q)(\ln p')^2} > 1.1 \tag{3}
$$
(A.Granville predicts that such counterexamples exist even for $q=2$, with the above ratio greater than $1.12$ -- more precisely, Granville expects that the ratio should exceed or come close to $2e^{-\gamma}$).
Question 4: Find a counterexample to $(2)$, with the additional condition
$p'-p>q^2$. 
Hint: Counterexamples are very rare. To find one, you will likely need to write a program and run it long enough. Good luck!

REPLY [2 votes]: Here are two counterexamples.
(A) Take $q=1605$, $r=341$, and consider the primes $p=3415781$ and $p'=3624431$. 
It is not difficult to check that 
$$
      p \equiv p' \equiv r \ ({\rm mod} \ q),  \tag{1}
$$
and between $p$ and $p'$ there are no other primes satisfying $(1)$.
We have $\varphi(1605)=848$,
and the exceptionally large gap is
$$ 3624431 - 3415781 = 208650 > \varphi(q) \cdot (\log3624431)^2 = 193434.64\ldots
$$
(This only answers question 1.)
(B) Take $q=18692$, $r=11567$, and consider the primes 
$p=190071823$ and $p'=193978451. \ $
We check that $(1)$ holds for $p$ and $p'$ -- and for no other primes between $p$ and $p'$. We have $\varphi(18692)=9344$; our exceptionally large gap is
$$
3906628 = 193978451 - 190071823 > \varphi(q) \cdot (\log193978451)^2 = 
3402811.2255\ldots
$$
(This answers questions 1 and 3.)
As of December 2019, questions 2 and 4 are still open.