This question is probably easy but I only have "tedious case checking" proof strategy in sight, and I'm sure there should be a reference lying around...

The question concerns the TSP problem (with fixed start- and end-vertex). The graph is the graph induced on the ball in an infinite grid. More precisely, the vertices are $B_n = \lbrace (x,y) \in \mathbb{Z}^2 \mid |x|+|y| \leq n \rbrace$ and two vertices $v'=(x',y')$ and $v=(x,y)$ are neighbours if $|x-x'| +|y-y'|=1$.

$\textbf{Questions:}$ For two distinct vertices $v$ and $v'$ in $B_n$ (say $n>2$) let $p_{v,v'}$ be the shortest path between these vertices which goes at least once through any vertex. Let $|p|$ be the length of a path $p$.

$\textbf{(1)}$ What is the smallest $k$ (possibly depending on $n$ but independent of $v$ and $v'$) such that $|p_{v,v'}| \leq |B_n| + k$? Can $k$ be chosen independently of $n$?

$\textbf{(2)}$ What is the smallest $\ell$ (possibly depending on $n$) such that $\max_{v,v'} |p_{v,v'}| \leq \min_{v,v'} |p_{v,v'}|+\ell$? Can $\ell$ be chosen independently of $n$?

For rectangles (of sides $m \times n$) instead of balls, it is known that $k$ and $\ell$ can be chosen to be (respectively) $1$ and $2$ as soon as $m,n \geq 3$. I have the impression that this is sharp contrast to the case of a ball ($\ell$ might still be uniformly bounded but $k$ is probably $\simeq An+B$)