Answer by Renat for AVX512BW: handle 64-bit mask in 32-bit code with bsf / tzcnt?

Looks like KSHIFTRQ could be used as an alternative, to right-shift top 32-bits of k0 counter to be lower 32-bits, which could be copied to the regular purpose register. Like:

.check_next_dword:
      add     eax, 32     
      KSHIFTRQ k0, k0, 32  ;shift hi 32 bits to be low 32 bits
      kmovd   ebx, k0   
    ...

And yes, vxorps zmm0, zmm0, zmm0 will set zmm0 to zero, as according to vxorps referense it's xor-ing without mask into 3-rd argument (you may check as well this SO question about zeroing zmm register)