Would DCBH form supermassive stars first before BH formation In reading some literature it seems ambiguous (intentionally?) as to the path that Direct Collapse Black Holes (DCBH) take. Some papers do talk about pristine Halos collapsing to form supermassive stars (SMS) ($10^{3}$) that then undergo direct collapse. However most that I've seen do not.
So my question is, do all DCBH models assume a short lived SMS phase or can there be a runaway collapse/accretion in the core of a H cloud that collapses into a intermediate mass black hole or larger? Are there multiple routes for DCBH? This is all assuming $z&gt;15$ and $\Lambda$CDM.
This article summarises the research:

How does a cloud of gas become a black hole? According to a 2017 study, such a transformation requires two galaxies with very different personalities: one of them a cosmic overachiever that's forming lots of baby stars and the other a low-key heap of starless gas.


As new stars form in the busy galaxy, they blast out a constant stream of hot radiation that washes over the neighboring galaxy, preventing the gas there from coalescing into stars of its own. Within a few hundred million years, that starless gas cloud could accrete so much matter that it simply collapses under its own weight, forming a black hole without ever producing a star, Basu found.

I assume that some fusion does happen during the collapse, but presumably it never catches up with the increasingly intense gravitational pull of the densening gas cloud, so never supports the gas against gravity