Let $p_1<p_2<\cdots$ be a sequence of primes such that $p_{i+1}\equiv 1\pmod{p_i}$. Is it true that

$$\lim_k p_k^{1/k}=\infty?$$

Does there exist such a sequence with

$$p_k \leq \exp(k(\log k)^{1+o(1)})?$$