A Costas array $\pi = d_1 d_2 \ldots d_n$ (one-line form) of order $n$ is a permutation $\pi \in S_n$ such that the $n-r$ differences
$d_{r+1} - d_{1} , d_{r+2} - d_{2} , \ldots , d_n - d_{n-r}$
are distinct for each $r$ , $1 \leq r \leq n-1$.
Find an example of a Costas array $\pi$ for some $n$ such that the $2n-3$ inequalities
$|d_2 - d_1| , |d_3 - d_2| , \ldots , |d_n - d_{n-1}| \geq 3$
and
$|d_3 - d_1| , |d_4 - d_2| , \ldots , |d_n - d_{n-2}| \geq 3$
hold or else prove that no such permutation exists.
Remarks: No such Costas arrays have been found using Beard's database of Costas arrays out to order $1030$ (the database is available at https://ieee-dataport.org/open-access/costas-arrays-and-enumeration-order-1030 and is exhaustive out to order $29$.) Costas arrays are known to exist for infinitely many (but not all) orders by constructions involving the infinitude of primes.