Volume 5, page 335, question 35 otherwise, question, interest, affect

yuoska

Volume 5, page 335, question 35
According to the answer:
“The value of a callable bond equals the value of an otherwise identical non-callable bond minus the value of the call. A rise in interest rates will reduce the value of the non-callable portion and thus reduce the value of the callable bond.”
Does not the rise in interest rates also affect the value of the call?
If we use a new interest rate tree with higher rates to value the call, we would get a lower call value.
The lower interest rates will cause the no callable to be worth less at any point in the tree, and thus less likely to be called, and thus call worth less…?