Are all connected graphs with degree sequence $(2,2,4,4,6)$ Hamiltonian?

Question

Are all connected graphs with degree sequence $(2,2,4,4,6)$ Hamiltonian?

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I have the following few observations:

Note that there are only $5$ vertices but the highest degree is $6$. Hence the graph is not simple.

It may contain loops or multi-edges. The vertices of degree $2$ can't contain any loops since the graph is connected.

Note that we can't apply Dirac's Theorem or Ore's Theorem since the graph is not simple.

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I tried finding an example but failed.

Need some help.