First, I don’t know the technical answer. I am not a Kant expert or a detail expert.
Whatever old metaphysics was, that’s what the things in themselves are. Loosely speaking a necessary structure.
I would use the same word as Hegel. A Logic. Hegel’s Logic is said to be Ontological. It’s the grand logical movements of Being that tie it all together and proceeds along, things like that. Even though I have always thought of Hegel as sort of open ended “until reason prevails” (my words).
You see the logical connections, purposes and ends especially with Aristotle and Aquinas. Hegel then is the new Aristotle-Aquinas-with Spinoza thrown in. Something like that.
Unmoved mover (Aristotle) sets it going because logically it has to do so, and there is a purpose to things. Or God (Aquinas) sets it going with ends in mind. Everything is connected.
What is NOT merely empirical. The big necessary logical narrative we arrive at through intuition as Klocking says. Philip helps me a lot with his discussion of intuition.
Kant gave us a restricted OS and software. Gauss and Einstein changed at least the Trans Aesthetic software. In other words these are software updates to Kant. Kant can be fixed with a mere software update perhaps, updating us with non-Euclidean geometry etc. Kant was a genius but he really was the end of “real” Philosophy. He knew this. He tipped his hat respectfully to things in themselves. Out of respect.
(The idea of making a kind of update to Kant was very generally E. Cassier’s idea)
Now we have philosophy as the mere handmaid to science.
Whereas the only real Philosophy is true, “logically required” metaphysics that we can know and discuss. You don’t just give a software update to a logically necessary Metaphysics.
Kant seems to say we can’t know big metaphysics, so Philosophers can’t discuss it. Then Hegel discusses it like crazy. Which of course changes the world (Marx etc) and makes Popper furious.
People are very wrong when they say we can just ignore Kant’s things in themselves. Kant was a brilliant by insisting on things in themselves. He retained that. He gives a kind of peculiar beacon of hope. A Hopeless hope and he showed respect.
Kant threw down the gauntlet, and Hegel eagerly picked it up. Maybe too eagerly. But Kant DID throw down the gauntlet, and we have to love him for it.
PS. The OP’s question seems to be an academic question within Kantian studies and I don’t know enough to answer it. I’m more interested in what the next major philosopher (Hegel) made of Kant’s project. Here is a review from Telos journal.
On Hegel’s Critique of the Noumenal in Kant Link https://www.telospress.com/on-hegels-critique-of-the-noumenal-in-kant/
Taking off from the above link:
There is no “One” that can stand apart. Meaning, in general, no Descartes. Which means we use Spinoza. Illustrated by Engels:
"Labor is the source of all wealth, the political economists assert... But it is even infinitely more than this. It is the prime basic condition for all human existence, ... in a sense, we have to say that labor created man himself" Friedrich Engels.
Ie our dialectical struggle with the world. Meaning real struggle. We can’t stand apart and consider our own cognition because it is built from an evolving (dialectical) encounter with the world. With Engels, “labor” We are immeshed in the process.
Another Engels quote: Quote: "So do you think", I asked, "old Spinoza was right when he said that thought and extension are nothing but two attributes of one and same substance?" "Of course", Engels replied, "old Spinoza was quite right". Conversation, Plekhanov and Engels.
And and also from the Telos review of Booth:
Kant preserved the “many” which man collapsed into one. Copernican Revolution Whereas in Hegel the many is already Forced into the One when man comes upon it. Concept of Force. “Hegel’s force overcomes the Kantian antinomy of the phenomenal and noumenal reality,”. See quote from Jean Hyppolite here:
When we envisage the fall of a body in space, we posit the same being twice: as reality, the motion is a juxtaposition that can be broken down into parts . . . but we can consider the “whole of the motion,” the integral of which it is the realization. We then have force, the content of which is identical to its manifestation, but which formally differs from that manifestation. Telos Hyppolite
PPS. Instead of Labor struggle, Hegel would suggest probably a Bildung of the Spirit. But it would still be a school of hard knocks.
The Telos article doesn’t use Engels, rather it features Adorno, Heidegger and Jean Hyppolite (Hegel Scholar) When I speak of Labor struggle here, I do not mean a struggle against Capital. But literally a struggle to survive in the world. Popularly speaking, evolutionary struggle.