In some inertial frame consider a disk of radius 1 lightyear at rest. Then along the edge of the disk there are some people in spacesuits at rest hovering right above the disk (which has negligible mass I really want it just to paint the scene).
There is a light source at rest in the center with a timer to turn it on and then off so that it will only ever be on for one single minute. When the people see the light they fire thrusters to get up to speed and move in a circle and then after that they fire their thrusters just to maintain uniform circular motion (again as observed by the frame of the disk). And then they hold hands.
When they stop seeing the light they stop holds hands but keep firing their tbrusters the same way. And then after they stop holding hands they fire their thrusters differently to get back to rest relative to the disk.
In the frame of the disk there was a connected rotating object of humans in spacesuits. But since they held hands for less than a minute and the disk is one light year apart (in the frame of the disk) the events of one person holding hands is spacelike separated from all the events where the opposite person held hands. So there are frames where all of the one events happen before all of the other events. And frames where all of the other events happen before all of the one events.
And in fact, if the speed they get up to is high enough, the instantaneously comoving frame of the person originally halfway between them is such a frame.
So whether there ever simultaneously existed a connected object consisting of mutually hand holding people is frame dependent.
There is a frame where this hand holding set of events is connected. And other frames (the frames that are instantaneously comoving with the parts) where the parts are never part of a single object.
I know that an object can be Born rigid if there is a frame where its components are at rest, e.g. from wikipedia
The defining property of Born rigidity is locally constant distance in the co-moving frame for all points of the body in question.
But this collection of components (people) do not have a common comoving inertial frame. An instantaneously comoving frame for one person is never an instantaneously comoving frame for the person initially at the opposite side for instance.
So there is no Born rigid object (except the disk which isn't involved, and the light source and the people but realistically every classical thing has parts that are moving so aren't Born rigid for the same reason, no comoving frame in common).
I fully understand that in one frame if something looks rigid we can call it Born rigid to accept that it might not look rigid to another frame. But that is only reasonable if there is a frame where the object has its parts all at rest. So this example isn't some connected object that is at rest in some frame, so we can't invariantly refer to the frame in which it is at rest because there is no such frame. And every single frame that has a part that is instantaneously at rest sees a component that is disconnected from the rest. Pun (about components) not intended.
So is being connected a frame dependent idea in Special Relativity given that the any instantaneously comoving frame of a component will observe itself as disconnected from other components?
And even if a frame sees some components connected to others, won't it have different shapes based on the spatial geometry of the connected components in the plane of simultaneity of the frame? Specifically that the object will be shorter in the direction the frame is moving relative to the disk compared to directions orthogonal to that motion? Again there isn't a common rest frame for the components to refer to.