Integers with a Hamiltonian Square Path

Question

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 that $\varphi(k)+\varphi(k+1)$ 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?

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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.

Answer 1

Post #22 at https://mersenneforum.org/showthread.php?p=477787 by R. Gerbicz claims a proof that $N=25$ is the answer, and that for $N\ge32$ there is a Hamiltonian cycle. See also the tabulation and discussion at the Online Encyclopedia of Integer Sequences.