Let’s consider a wire strung between two posts.  After you hit it with a hammer two waves pulses propagate up and down the string, one in each direction. They hit the end posts, bounce in the other direction, and so on; two pulses racing back and forth along the length of the string.

The fundamental frequency, ie the pitch of the string, is the inverse of of the round trip time.

First, consider the case where the internal friction of the string is very low. Then the losses occur when the pluses hit the end posts.  For higher frequency vibrations, this happens more rapidly.  Assuming that the same fraction of the wave energy is lost from the string (and transferred to the sound board) each time, you’d expect the sound to dissipate more rapidly for higher frequencies.

Similar considerations apply when thinking about internal losses — a given short segment of the string flexes and then flattens as the pulse moves through it.  And again, if each flex dissipates some energy, then more energy is lost per unit time for higher frequency strings.

So, to first order, keeping everything other than the length of the string fixed, you’d expect that higher pitched strings would lose energy more rapidly than lower pitched ones.  

Of course, in a real piano not everything else is held fixed, and then when you start to consider the fact that the perception of loudness depends on frequency, then things getting more complicated fast.