How to get a speedup linear in the number of iterations on unsalted iterated password hashing?
The only way I see to justify the claim is that the authors are thinking as follows thus it is not for any $P'\neq P,$ but specific $P'$ in that trail:

Compute $H^i(P),$ for $i=1,2,\ldots,k.$ This gives you the hash for $P,$ call it $H_0.$

Now define the sequence $P'_i=H^i(P),$ for $i\geq 1,2,3,\ldots.$

Evaluate $H^i(H_0)$ to obtain $H(P'_i),$ for $i\geq 1,2,3,\ldots,$ at extra cost of one evaluation each. Of course these $P_i'$ are almost certainly distinct until the trail length is comparable to square root of the hash output space.