"The states have submitted more than 500 requests for such a convention to Congress, with the required two-thirds of the states asking for such conventions. The state applications for an Article V convention are registered in the Congressional Record. Congress has violated the Constitution by ignoring these requests. In so doing, Congress has destroyed one of the checks on federal power that the founders had implemented."
If the states agree on a "general" convention, that's what would occur. But only three times in our history, all by states who were unhappy with the results from Philadelphia, have states sought a general convention.
All other conventions must be specific to the agreed subject. And a call for an amendment to get rid of the UN does not match in subject matter a call for a balanced budget amendment. And Congress is not obligated under Article v to call such a convention until 34 states have agreed on the purpose of that convention.
No such agreement has ever been reached. The article you cite ducks this question, which is decisive.
(Congressman Billybob, I may need some legal principles enunciated here.)
Congress, from the beginning, chose not to recognize petitions on different subjects enacted at wildly different times. There were two unwritten standards: "single subject" and contemporaneousness. In 1992, Congress finally legislated both concepts into law.
Those two standards -- first unwritten and then later enacted into law -- permitted Congress to collate petitions by single subject and then weigh their timeliness. Part of this is connected with the legal concept of "agency".
The Convention is an agent of the states, and in their petitioning language, the states lay out the issue(s) on which they deputize the Convention to act as their agent. An agent cannot act outside the authority granted him. Under basic concepts of law, the Convention cannot stray from the mandate given it by the states. This is the source of the "single subject" condition.
The commotion over the ratification of 27th Amendment caused Congress, in that 1992 law, to limit ratifications of an amendment by the states to a 7 year shelf life. Congress applied the same rule to petitions for a Convention for Proposing Amendments.
In 2000 Walker v. US attempted to litigate the issue of the petitions for a convention that had been piling up for 200+ years. A federal court in the 9th District dismissed the case.