What complexity class does this variation of traveling salesman problem belong to?

Given a TSP instance $T$, decide whether changing the city coordinates by adding a vector of coordinates $v$ will change the optimal TSP objective by atleast $x$. The city coordinates are integers.

The problem is in PSPACE but even the verification problem seems to be NP-hard. Is that true?

If the verification problem is NP-hard, what exact complexity class does this problem belong to?

I'm assuming that you're thinking of the Euclidean traveling salesman problem, where $c$ and $v$ are vectors in $\mathbb{Z}^{2n}$, with two coordinates for each city. Let $minTSP(c)$ denote the length of the minimum traveling salesman tour for the cities with coordinates $c$. Then your problem asks whether
$$
minTSP(c + v) \ge min TSP(c) + x.
$$
But then the special case $c=0$ is equivalent to asking whether $minTSP(v) \ge x$, which is coNP-hard.

StackExchange Computer Science Q#2737, answer score: 10