The EKG linear-bound conjecture

Let $(a(n))_{n\geq 1}$ be the EKG sequence defined by $a(1)=1$, $a(2)=2$, and, for $n\geq 3$, choosing the smallest unused natural number sharing a nontrivial common divisor with $a(n-1)$. The EKG linear-bound conjecture. The sequence satisfies

$$a(n)\geq\frac{13}{28}n,$$

with equality if and only if $n=28$, and

$$a(n)\leq\frac{12}{7}n,$$

with equality if and only if $n=7$. These sharper bounds are supported by numerical evidence, whereas the paper proves only substantially weaker linear bounds.