The prime test takes a specific amount of iterations that is proportional to the log of the tested number (i.e. the exponent). However, each iteration takes longer if the FFT size is large as in that case more multiplication steps are required per iteration. If you run llr2 manually you'll see it prints a "time per iteration", that increases with increasing FFT size.
The FFT size depends on both the size of the number and the size of k. And k=3 is the smallest one can get. You'll see that computation times for the different k's in the conjecture subprojects vary strongly depending on the respective k.
A short and more mathematical explanation can be found here:
As to why a large FFT requires more multiplications per iteration, I have no idea. Maybe some of the more knowledgeable members can explain that. :)
1281979 * 2^485014 + 1 is prime ... no further hits up to: n = 5,600,000