Legendre's conjectural prime recursion for a Mills-type construction

Define $q_1=2$, and for each positive integer $n$, let $q_{n+1}$ be the least prime exceeding $q_n^2$. Legendre's recursive prime-gap assertion. Then

$$q_{n+1}<(q_n+1)^2$$

for all $n$. This assertion is the instance of Legendre's conjecture needed for the displayed recursive construction of a real number whose powers yield primes; the paper also notes that a related existence result is known unconditionally by work of Kaisa Matomäki.