Open mathematical questions for which Numerical Search promising

(Posting this as an answer, in part because the comments thread is already too long.) Here is a relatively new open question (2016) where a numerical search for counterexamples seems promising:

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 a residue class: $$ p'-p ~<~ \varphi(q) \log^2 p'. \tag{2} $$

(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. As usual, here $\varphi(n)$ denotes Euler's totient function.

Note: The logarithm in $(2)$ is taken of the prime $p'$ at the larger end of the "gap" $[p,p']$. (If instead we take the log of $p$, the smaller end of the "gap", then counterexamples are easy to find.)