Non-adaptive simulation impossibility conjecture for irregular communication orders

Let $Pi$ be an interactive protocol with a sufficiently non-regular, for example pseudo-random, communication order, and let $Pi'$ be a non-adaptive protocol whose communication order is fixed in advance. Let $R$ denote the simulation rate. Non-adaptive simulation impossibility conjecture. Such a protocol $Pi$ cannot be robustly simulated by any non-adaptive protocol $Pi'$ with rate

$$R=1-o(1).$$

This should hold for essentially any channel introducing some error, including a channel introducing only a single random error or erasure. The conjecture concerns the strong limitation imposed by non-adaptive communication orders, in contrast with adaptive simulations.