Is there a connection between Burness and Campbell of Argyll?

We have been having an email conversation with someone looking into whether Burness is indeed associated with Cambell of Argyll. In Books such as 'Tartan for Me' it does link Burness with Campbell, but we have seen no written evidence to prove this. So where does it come from?

John Burness from The One Name Study 'Burness Geneaology and Familly History' has commented:

"There is no truth to the supposed connection between the Burness family and Campbell. The story originated with an application by Sir James Burnes for a coat of arms, in which he included the Campbell arms based on an account he received from a cousin who had heard the story from an elderly Burness many years previous to that. Upon further investigation of his family Sir James found that the "facts" in this story couldn't be true and applied for a new coat of arms omitting the Campbell arms."

It would be interesting to hear if anyone else has done any research in this area.