Consider the following input:

In[1]:= c = 1.2``200; n = 7;
2/Pi NIntegrate[ChebyshevU[n, x]^2 Sqrt[1 - x^2], {x, -1, 1}, 
  WorkingPrecision -> 200]
(2 I)/(c^2 Pi) 
 NIntegrate[ ChebyshevU[n, k/(I c)]^2 Sqrt[c^2 + k^2], {k, I c, -I c},
   WorkingPrecision -> 200]

Out[2]= 1.\
0000000000000000000000000000000000000000000000000000000000000000000000\
0000000000000000000000000000000000000000000000000000000000000000000000\
000000000000000000000000000000000000000000000000000000000000

During evaluation of In[1]:= NIntegrate::precw: The precision of the \
argument function \
(\[Sqrt](1.\
4400000000000000000000000000000000000000000000000000000000000000000000\
0000000000000000000000000000000000000000000000000000000000000000000000\
00000000000000000000000000000000000000000000000000000000000+k^2) \
((0.+<<18>> \[ImaginaryI]) k+<<2>>+(0.+<<1>>) <<1>>)^2) is less than \
WorkingPrecision (200.`). >>

Out[3]= 1.\
0000000000000000838554115034660068008817729444357599619488431719194210\
2454049745574593544006347656250000000000000000000000000000000000000000\
00000000000000000000000000000000000000000000000000000000000

These two integrals are related by a simple transformation. Why then the first integral is calculated with the prescribed accuracy, and the second is not? What to change in the code so that the second integral would also be computed with accuracy 200 digits?