Yes. We prove that there exists a sequence $(x_1,x_2,\ldots,x_n)$ of any length at least 2 satisfying ${lcm(x_{k-1},x_k}) > lcm(x_k,x_{k+1})$. Firstly, when $n=2$ there is nothing to prove. If $n\geq2$ take a sequence $x_1,x_2,\ldots,x_n$ satisfying the properties. Now note that $dx_1,dx_2,\ldots,dx_n$ also satisfies the properties for any $d$, so for $d$ large enough such that $d \cdot lcm(x_1,x_2) < dx_1(dx_1-1)$, we can take $dx_1-1,dx_1,dx_2,\ldots,dx_n$ as the new sequence of length $n+1$. Thus by induction we are done.