Cloitre's prime-or-one conjecture for the least-common-multiple sequence

Define

$$a_n=\begin{cases}1&\text{for }n=1,\\a_{n-1}+\operatorname{lcm}(n,a_{n-1})&\text{for }n\geq2,\end{cases}$$

and, for $n\geq2$, define $b_n=a_n/a_{n-1}-1$. The sequence $(b_n)$ consists of positive integers. Cloitre's conjecture. For any $n\geq2$, $b_n$ is either $1$ or a prime number. Numerical evidence supports the conjecture, but the paper states that no proof is known; a sufficient condition is related to Linnik's theorem.