I take that to contain two points: (1) holding a cloture vote would overturn Senate precedent and establish that 60 votes are needed to confirm a judge; and (2) holding a cloture vote would unconstitutionally change the meaning of "advise and consent."
On point (1), I think the Democrats have already set the precedent by mounting the filibuster. I think failing to hold the cloture vote and thereby admitting defeat would much more serve to consolidate that precedent than holding a cloture vote, hopefully followed by further debate and cloture votes.
On point (2), if the Constitution's "advice and consent" language doesn't permit filibusters, then the filibuster of Abe Fortas in 1968 was unconstitutional. I've never heard anybody argue that. I don't think it's a plausible view. I would be willing to bet that courts would reject the argument. I don't believe such a view of the Constitution is consistent with allowing filibusters of legislation, which I believe the courts have upheld.
Maybe I'm being anal here (it won't be the first time), but is a filibuster not a filibuster until it is attempted to shut it off? The House has a cloture rule that automatically ends debate; the Senate has a cloture vote that allows for unlimited debate. Until someone tries to stop debate, it is just debate. Once an attempt to stop debate fails and debate continues, wouldn't that then become filibuster?
On point (2), if the Constitution's "advice and consent" language doesn't permit filibusters, then the filibuster of Abe Fortas in 1968 was unconstitutional. I've never heard anybody argue that.
I looked up Foras' brief bio on an on-line almanac site, and I think the differentiation is this: Fortas was already on the Supreme Court. The filibuster was on his appointment as Chief Justice.