This is the way I saw it, and it yielded me to the right answer (somehow).
The price of the putable bond is equal to… Putable Bond = Straight Bond + Put Option
An increase in the yield curve, makes the bond more risky to a price decline. Therefore, in a riskier bond the price of a put option is higher. (kind of like for a more volatile stock, the price of its options increase).
Since the straight bond didn’t change in price, then at least the option must reflect the higher risk. Therefore, the option price increased.
Good luck and I hope it helps.