(This is a natural counterpart to question 3132001 which deals with upper bounds.)
Let $q$ and $r$ be coprime integers, $1\le r < q$, and consider the arithmetic progression $$ r, \ r+q, \ r+2q, \ r+3q, \ldots \tag{P} $$
Dirichlet proved that there are infinitely many primes in progression (P) when $\gcd(q,r)=1$. This is Dirichlet's theorem on arithmetic progressions.
Let $R(n,q,r)$ be the $n$th record gap between primes in progression (P). For example, with $q=6$ and $r=5$, we have $R(n,6,5)=\mbox{A268928}(n)$; see http://oeis.org/A268928.
Conjecture: Almost all record gaps $R(n,q,r)$ satisfy $$ R(n,q,r) > {1\over6} \varphi(q) n^2. \tag{L} $$ Here $\varphi(q)$ is Euler's totient function.
Question: Find counterexamples to inequality (L). (Please find as many as you can. You will likely need to write a program and run it long enough.)
