Erdős–Selfridge prime classes: infinitely many primes per class, and growth of $p_r^{1/r}$ (Erdős #1055)

Consider the following classification of the primes, due to Erdős and Selfridge. A prime $p$ is said to be in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. Recursively, for $r\geq 2$, a prime $p$ is in class $r$ if every prime factor of $p+1$ lies in some class $\leq r-1$, with at least one prime factor lying in class exactly $r-1$. Two questions. (1) Are there infinitely many primes in each class $r$? (2) If $p_r$ denotes the least prime in class $r$, how does $p_r^{1/r}$ behave as $r\to\infty$ — does it tend to infinity, or remain bounded?

Acceptance. FULLY RESOLVES: a complete proof (machine-checkable preferred, else full written proof) that there are infinitely many primes in every class $r$, together with a proof determining the asymptotic behaviour of $p_r^{1/r}$ — either $p_r^{1/r}\to\infty$ (Erdős) or $p_r^{1/r}$ bounded (Selfridge). ADVANCES: extend the verified sequence $p_r$ of least primes in class $r$ (OEIS A005113) beyond its currently known terms, with the search program and a reproducible certificate that each new $p_r$ is genuinely the least prime whose $p+1$ has the required recursive class structure (and that no smaller prime qualifies); prove infinitude of primes for a specific new class $r$; or establish rigorous bounds on $p_r$ as a function of $r$ that constrain $p_r^{1/r}$ (a one-sided unconditional bound settling boundedness or unboundedness in part). Deliver the proof, or the class-certifying search code plus the extended sequence and its exhaustiveness certificate.