reference request - Is isobutyl chloride or ethyl chloride more reactive in an SN1 reaction?

Compare the rate of $\mathrm{S_N1}$ reaction in the following compounds:

We see that both compounds form primary carbocations. However, there is more hyperconjugation in the second compound as compared to the first one.

Also as the step in which the leaving group leaves is rate determining step (slowest step), which is before the rearrangement of the carbocation, I concluded that the second compound must have a greater rate of $\mathrm{S_N1}$ as compared to the first one.

But the answer says that the first has a greater reactivity than the second. Is the answer correct? If yes, then why?