The principle of (special theory) relativity states that the laws of physics should have the same form in all inertial frames of reference.
So if K.E. of particles has to be conserved for relativity to hold, then relativity will not apply to inelastic collisions (the type where two objects stick together after collision) . If this condition is a must relativity, would be a severely limited theory which is not the case. So this cannot be a condition. Note that momentum is always conserved even in the above case.
Now coming to the free particle part. A free particle has no net force acting on it and hence no net acceleration. This also cannot be a condition as we would not be able to study accelerated motion of particles
The condition is only that the frames of reference be inertial i.e. they themselves do not accelerate. A particle within such an inertial frame can accelerate.
To know what happens with accelerated frames one needs the general thoery of relativity.
Maybe the question should have been worded differently.