Mechanism of acid-catalyzed ring opening of a cyclopropane ring

I have a question regarding following synthesis which apparently only needs a sour medium to occur:
[![enter image description here](https://i.stack.imgur.com/bOLdo.png)](https://i.stack.imgur.com/bOLdo.png)


What exactly happens as a mechanism? What I thought was: The alcohol group gets protonated and forms a stable carbocation at it's tertiary position. This charge can now wander to the tertiary carbon next to it; but an H-Shift should not be possible. What is the mechanism here?




I believe the mechanism should be somewhat like this (after the formation of carbocation) -


[![enter image description here](https://i.stack.imgur.com/JuEO8.png)](https://i.stack.imgur.com/JuEO8.png)


It would account for the product shown, but I am not sure if it's correct. I feel it's more favourable for a dehydration to occur with the hydrogen from the ring itself (shown on the dash at the top) to form this -


[![enter image description here](https://i.stack.imgur.com/WsUNZ.png)](https://i.stack.imgur.com/WsUNZ.png)


This double bond is conjugated and more substituted, so I should have thought that this would be the major product.


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