Inverted Yield Curve and Embedded Option

Why does the value of a put option decrease if the yield curve inverts from an upward sloping yield curve?

Is it because as the bond gets closer to maturity, and we enter the inverted yield curve phase, the price of the putable bond goes up and the put option is out of the money?

Thank you!

I think you’re right. Lower rates will lead to higher bond prices, thus making it less likely that the put option will be exercised.