the paper "In defense of the Simplex Algorithm's worst-case behavior" Disser/Skutella [1] was recently cited on this tcs.se site by saeed on another interesting question. the paper introduces the idea of "NP mighty" algorithms (p3 def2). it follows a fruitful/continuing line of research analyzing P$\stackrel{?}{=}$NP wrt the simplex algorithm and linear programming of which there have been other major recent advances, eg results by Pokutta et al in [2] showing that the P-time TSP polytope must have an "unlikely/restrictive" form (commentaries in Barriers to P/NP proofs, RJLipton, also Stating P$\stackrel{?}{=}$NP without TMs). question (possibly with multiple leading answers):
the Disser/Skutella paper has closely related ideas but does not seem to explicitly reformulate the P$\stackrel{?}{=}$NP question. what is an equivalent way to state/study it in their introduced schema/framework of "NP Mighty" algorithms? what is a/the basic open problem in simplex/linear programming complexity theory that is equivalent to P$\stackrel{?}{=}$NP?
(somewhat related question: the Disser/Skutella paper also refers to Klee-Minty cubes, long used to show worst-case behavior on the simplex algorithm. are there any results relating lower bounds on them to general algorithmic lower bounds and/or complexity class separations eg P$\stackrel{?}{=}$NP etc?)
[1] "In defense of the Simplex Algorithm's worst-case behavior" Dissker/Skutella
[2] Exponential Lower Bounds for Polytopes in Combinatorial Optimization Fiorini et al