What do you mean by "the tone will die off"?
As pointed out before, new strings tend to sound crisp and bright, at the beginning.
Usually, after a while they settle in; sound less bright and crisp - it's a matter of taste what's the intended string sound;
A: strings XYZ sound good after the break-in phase;
B: strings XYZ only sound good from the start - until they change.
C: don't care
In case of A, the break-in period is the time you may want to play with the tone knob; to take away the excess brightness.
In case of B, and since the break-in phase is usually only a fraction of a string's lifecycle, buying new strings regularly is not within everyone's budget. So, if one likes/needs the brighter sound, it may be worth trying nanocoated strings, such as Elixir roundwounds which indeed keep their brightness long; I am using mine since last year's summer, and comparing the practice recordings, they still sound the same. Though, my Rotosound still sound the same since maybe the 70s - and hopefully much longer
Just saying, it depends what
you want / need.