For a prime $q_k \neq 2$ we can study the corresponding set $\{q_1!+q_k,...,q_{k-1}!+q_k\}$, where $q_1,...q_{k-1}$ are all primes strictly less than the prime $q_k$.

Peter and Mathphile computed that for $q_k=1193$ we have that all the numbers from the corresponding set are composites

I think that $1193$ is rather large as an example of the first prime that generates all composites in the corresponding set and I do not see some reasons, armed with strong principles, of why there shouldn´t be some other primes that generate corresponding sets in which not a single number is prime.

Suppose that $\{r_1,...,r_m,...\}$ is the set of all the prime numbers for which the corresponding sets consist of only the composite numbers.

Are there any reasons and principles for justifying the assertion that $\{r_1,...,r_m,...\}$ is not a finite set?

Edit: This question arrived as the result of thinking about Mathphile´s conjecture.