Are there an infinite number of prime quadruples of the form $10n + 1$, $10n + 3$, $10n + 7$, $10n + 9$?

In base 10, any prime number greater than 5 must end with the digits $1$, $3$, $7$, or $9$. For some $n$, $10n + 1$, $10n + 3$, $10n + 7$, $10n + 9$ are all prime: for example, when $n=1$, we have that $11$, $13$, $17$, and $19$ are all prime. My question is, can anyone disprove the claim that there are an infinite number of such primes. (If you can prove it, that is fine, too, but seeing as it is a stronger form of the twin primes conjecture, I’d imagine that would be difficult).

The only progress I’ve been able to make is to show that $n$ must be of the form $3k + 1$ by considering the system of modular inequalities
$$p \not\equiv 0 \mod{2}$$
$$p \not\equiv 0 \mod{3}$$
$$p \not\equiv 0 \mod{5}.$$

Beyond that, I don’t know where to go.

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