Can we construct a Karp reduction from a Cook reduction between NP problems?

We have had several questions about the relation of Cook and Karp reductions. It's clear that Cook reductions (polynomial-time Turing reductions) do not define the same notion of NP-completeness as Karp reductions (polynomial-time many-one reductions), which are usually used. In particular, Cook reductions can not separate NP from co-NP even if P $\neq$ NP. So we should not use Cook reductions in typical reduction proofs.

Now, students found a peer-reviewed work [1] that uses a Cook-reduction for showing that a problem is NP-hard. I did not give them full score for the reduction they took from there, but I wonder.

Since Cook reductions do define a similar notion of hardness as Karp reductions, I feel they should be able to separate P from NPC resp. co-NPC, assuming P $\neq$ NP. In particular, (something like) the following should be true:

$\qquad\displaystyle L_1 \in \mathrm{NP}, L_2 \in \mathrm{NPC}_{\mathrm{Karp}}, L_2 \leq_{\mathrm{Cook}} L_1 \implies L_1 \in \mathrm{NPC}_{\mathrm{Karp}}$.

The important nugget is that $L_1 \in \mathrm{NP}$ so above noted insensitivity is circumvented. We now "know" -- by definition of NPC -- that $L_2 \leq_{\mathrm{Karp}} L_1$.

As has been noted by Vor, it's not that easy (notation adapted):


Suppose that $L_1 \in \mathrm{NPC}_{\mathrm{Cook}}$, then by definition, for all languages $L_2 \in \mathrm{NPC}_{\mathrm{Karp}} \subseteq \mathrm{NP}$ we have $L_2 \leq_{\mathrm{Cook}} L_1$; and if the above implication is true then $L_1 \in \mathrm{NPC}_{\mathrm{Karp}}$ and thus $\mathrm{NPC}_{\mathrm{Karp}} = \mathrm{NPC}_{\mathrm{Cook}}$ which is still an open question.

There may be other differences between the two NPCs but co-NP.

Failing that, are there any known (non-trivial) criteria for when having a Cook-reduction implies Karp-NP-hardness, i.e. do we know predicates $P$ with

$\qquad\displaystyle L_2 \in \mathrm{NPC}_{\mathrm{Karp}}, L_2 \leq_{\mathrm{Cook}} L_1, P(L_1,L_2) \implies L_1 \in \mathrm{NPC}_{\mathrm{Karp}}$?

its a generally open TCS problem subject to ongoing research whether & the exact conditions Cook & Karp reductions are equivalent and is apparently closely related to the open NP=?coNP question & other complexity class separations eg E=?NE (wrt sparse languages).

here are two research papers on the subject & further leads on tcs.se via similar question:

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Are Cook and Karp Ever the Same? Beigel, Fortnow

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Cook vs. Karp-Levin: Seperating Completeness Notions If NP Is Not Small (1995) Lutz, Mayordomo

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many one reductions vs Turing reductions to define NPC, tcs.se

StackExchange Computer Science Q#20074, answer score: 4