Ring expansion in cyclo compounds In the following reaction, why does ring expansion not take place?

Mechanism that was the correct one:

In the second step when there is a protonated carbon the ring should expand to become 6 membered as that is more stable. This should be followed by a methyl shift from Quaternary to tertiary and then the $\ce{Cl}$ should get attached. But this doesn't happen. Why is that? According to my textbook's theory, what I described should occur but the mock test says otherwise. Are there some more criteria-besides.

protonated carbon next to ring,

forming of a more stable 6 number ring


for ring expansion that my text book is missing?
Any help would be appreciated.
Simple answer is consider the stability of carbocations. You already got the tertiary carbocation in hand that is much stable than secondary carbocation, which would be resulted by ring expantion. Sure, a methide migration would give you another tertiary carbocation, but before you get that, the reaction progress has to climb another actvation barrier, which would be steep ($3^\circ \rightarrow 2^\circ$). Bottom line is rate of clorination of carbocation is faster than rate of this rearrangement followed by methide migration ($E_\mathrm{a2} \lt E_\mathrm{a3}$):