Cloitre's conjecture on the primes occurring in the least-common-multiple sequence

Let $a_n$ and $b_n$ be defined by

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

and $b_n=a_n/a_{n-1}-1$ for $n\geq2$. Cloitre's prime-occurrence conjecture. The sequence $(b_n)$ contains every prime number other than $3$, which never appears. The paper says this conjecture is proved in its Section 3, using two propositions; in particular, for every prime distinct from $3$, $b_p=p$, and the paper establishes that $3$ does not occur.