The EKG prime-neighbor conjecture

Let $(a(n))_{n\geq 1}$ be the EKG sequence defined by $a(1)=1$, $a(2)=2$, and, for $n\geq 3$, letting $a(n)$ be the smallest natural number not already in the sequence such that $\operatorname{gcd}(a(n-1),a(n))>1$. The EKG prime-neighbor conjecture. Whenever a prime $p$ occurs in the sequence, it is immediately preceded by $2p$ and hence followed by $3p$. This is suggested by numerical computation through the first $10^7$ terms, but the claim is not proved.