Acid Catalysed Ring Expansion – Mechanism?

Ring expansion of vinyl cyclobutane - 


[![enter image description here](https://i.stack.imgur.com/3UWbq.jpg)](https://i.stack.imgur.com/3UWbq.jpg)


Could someone please help me with a detailed mechanism for the above conversion? 


It is the acid-catalyzed rearrangement of 4,7,7-trimethylbicyclo[3.2.0]hept-3-en-6-one to 3,3-dimethyl-6-methylidenebicyclo[2.2.1]heptan-2-one.


I couldn't find the mechanism anywhere, and couldn't propose a good one either.
Please help. 


P.S.
As one of the users in the comments is interested in seeing my best effort, I'll explain my thought process-
As the carbonyl oxygen is the most nucleophilic, it is the first to be protonated in acidic medium. Furthermore, this species has a resonating structure (less contributing, but the only one that can lead to a ring expansion, I believe) – one obtained by shifting the π bond towards oxygen – a carbocation. 
After this, I'm clueless. I think, that the expansion could have something to do with the hydrogen at alpha position to the sp² hybridised carbon atom in the five membered ring – as the product has an exocyclic double bond. 




You are correct about protonating the carbonyl group. [The first of your products is the same as the reactant.] The cyclobutanone **2** was prepared from a mixture of geranic acids. ([J. J. Beereboom, *J. Org. Chem.*, **1965**, *30*, 4230](https://pubs.acs.org/doi/pdf/10.1021/jo01023a056)). The structures of **2**, **4** and **5** were determined by Beereboom. I am providing the most reasonable mechanism. Cyclobutanone **2** must have a more stable cis ring juncture owing to its method of formation or exposure to p-TsOH. The *trans* isomer of **2** cannot give the *trans* isomer of **3** because of strain. Tertiary carbocation **3** arises by addition of the double bond to the protonated carbonyl. Collapse of the carbocation produces bicyclic ketone **4**, which in the presence of p-TsOH isomerizes to the exocyclic isomer **5**. In general, 1,1-disubstituted double bonds are less stable than tri-substituted ones. In this case, ring strain may be present in the endocyclic alkene **4**. 


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


 5