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The ABA Report lays out the reasons why the commissioners who wrote the report believe that delegates to an Amendments Convention would be elected by the people of the states on a "One Man/One Vote" basis. The reason is Reynolds v. Sims and their belief that it would govern an Amendments Convention. One other reason is that the original Convention of 1787 was held under the rules of the Articles of Confederation, and those rules were superseded by the Constitution.

I would recommend printing off both reports to which I linked and reading them carefully.

In the end, Congress can do certain things to blunt the efforts of the states:

Congress can attempt to refuse to call a convention. ALEC thinks that the courts would get involved and use the old Powell decision to tell Congress to do its duty. Even ABA sees that as a likely outcome. Others believe that the courts would view it as a political question and let the people remove recalcitrant senators and congressmen in elections if they truly want an Amendments Convention.
Congress can attempt to regulate the Convention. ABA believes this is a prerogative of Congress and lays out the arguments under the Dillon and Coleman decisions as to why this is so. ALEC believes that the states own and operate the Convention, and Congress has no authority to do anything but set the time and place of the Convention.
Congress can create its own weak amendments trimming its power and refuse to call a convention. This is how Congress weaseled out of a convention in 1913 over the 17th Amendment. But Congress had a legal technicality in that case. Several states had said in their petitions for a convention that if Congress passed an amendment to the states permitting the direct election of senators, their petitions would be considered as discharged. Once Congress did that, those few petitions were considered discharged, and the two-thirds threshold was no longer met.

So how do I think it will play out?

If 34 or more states can agree on identical language, Congress will not have a legal leg to stand on, and rather than let the federal courts get involved, Congress will swallow hard and set the time and place of the Amendments Convention.
Congress will then resurrect Orrin Hatch's bill of 1991 that would place strict rules on the Convention. It would be best if it doesn't pass. But if it passes, the states will go to federal court to argue that Congress has overstepped its bounds by getting involved in a state matter.
The Supreme Court will either add on to the Dillon and Coleman decisions by letting the new law stand, or it will throw the law out and let the states run things. It will be a 5-4 decision either way.

There have been other movements to convene the states for the purpose of amendments.

Each time, congress stepped in to sidetrack the process.

There is no doubt congress will attempt the same.

I expect rat and rino skulduggery behind the scenes. Individual state legislators will be cajoled/bribed/threatened.

If that does not work, congress will attempt to compromise the amendments, by offering their own watered down versions.