From Kozen INDEXINGS OF SUBRECURSIVE CLASSES: "the class of polynomial time computable functions is often indexed by Turing machines with polynomial time counters.... The collection of all (encodings over (0,1)) of such machines provides an indexing of 𝑃𝑇𝐼𝑀𝐸...we have Theorem: No universal simulator for this indexing can run in 𝑃𝑆𝑃𝐴𝐶𝐸."
He then goes on to say: "can it be proved that 𝑔𝑟𝑈 for any indexing of 𝑃𝑇𝐼𝑀𝐸 requires more than 𝑃𝑆𝑃𝐴𝐶𝐸 to compute? We have proved this for a wide class of indexings, namely counter indexings satisfying the succinct composition property."
𝑔𝑟𝑈 is the graph of the universal function 𝑈 and represents the minimum power necessary to simulate 𝑃 uniformly.
The counter indexing (or polynomial time counters) he is referring to is talked about in the answer here: https://cs.stackexchange.com/questions/126615/how-does-an-enumerator-for-machines-for-languages-work/126621?noredirect=1#comment266467_126621
I'm wondering what causes simulation of the enumerated machines to be difficult, and can it be circumvented--namely: does there exist an enumeration of machines for all languages in $𝑃$ that describes the machines in such a way that allows the machines, as specified, to be simulated efficiently by a simulator (with no more than 𝑃𝑇𝐼𝑀𝐸 overhead) on any input--or (basically) as Kozen puts it "allow(s) easy construction of programs from (their) specification"?
Looking for any progress that has been made since Kozen asked ^ the above question ^.