Option-embedded bonds Value when YC Flattens

I just read the following note: “Value of option-embedded bonds DECLINES as an upward sloping yield curve flattens.”

I don’t understand this. Can someone explain?

First of all, my understanding is that flattening YC means that rates are going down, is that correct?
Then the effect of rates going down should be different on callable and putable bonds, so how could it be that BOTH callable and putable bonds decline in value?