In his 1985 paper, Kyle presents 3 versions of the same model: a single period model, a multiple period model and the continuous time limit of the multiple period model.
When he formalizes the equilbrium problem for the discrete time multiple period model, he restricts himself to recursive linear pricing (P) and demand (X) rules. He writes on page 1322: " We suspect, but have not been able to prove, that equilibria with nonlinear X and P do not exist."
Has anyone ever proved or disproved this conjecture? Is it a redundant restriction as he suspected, or is it something binding and there are equilibria that would yield greater profit to the insider, albeit with nonlinear demand schedules?
If this appears to remain an open question, is there any relatively recent reference that says we're still unclear about this?
Thanks in advance.