Let $n>1$ be an integer and set $[n]=\{1,\ldots,n\}$. We say that $n$ has a "Hamiltonian Square Path" if there is a bijection $\varphi:[n]\to[n]$ such that for all $k\in [n-1]$ we have $\varphi(k)+\varphi(k+1)$ is a square number.

For instance $15$ and $16$ have this property.

Question. Is there an integer $N>1$ such that every integer $n\geq N$ has a Hamiltonian Square Path?

Note. This problem can be formulated in the language of graph theory and Hamiltonian paths. We say that $a\neq b\in [n]$ form an edge if their sum is square, and the above question is about integers such that the resulting graph has a Hamiltonian path.