Total Variation of a Signal - Is It Proportional to Signal Energy?
No it is not.
Total Variation is like the amount of changes in the signal.
Though changes require energy it doesn't mean they are proportional.
For instance, imagine that during a Window we see a constant signal of high value.
Clearly this high energy signal (Unless energy for you is the Variance, usually it is the 2nd moment) yet its Total Variation is zero.
For instance:

While the green signal has low TV but high energy the red signal has higher TV yet smaller energy.