High-density flux conjecture for TASEP-H with I-perturbations

Let $\rho$ be the particle density, let $h$ be the critical density, and let $\phi^*$ denote the maximal-equilibrium-state flux. Consider TASEP-H with I-perturbations. High-density flux conjecture. If $h<\rho<1/2$, then

$$\phi(\rho)=\phi^*.$$

This conjecture asserts that I-perturbations have the same high-density limiting flux as A-perturbations; the source presents it as a natural conjectural extension of the proved A-perturbation result.