My impulse response does not tend to zero
Note that the DFT transforms a periodic sequence into a periodic sequence. Hence, the result of your IFFT is intrinsically periodic and you should view it as such. The parts you are seeing at the end of it are thus just as well behind the start as they are before it. 

Hence, this part may as well be some form of pre-ringing, stemming from a nonzero group delay of your filter. 

Note also that the FFT implicitly periodifies your spectrum. It looks a little cut, so you may be missing some essential parts.