Threshold conjecture for the scaling limit of TASEP with a long caterpillar

In the setting of the section, consider right Bernoulli TASEP with sequential updates and an additional caterpillar of length $b\varepsilon^{-1/2}$ placed at distance $b'\varepsilon^{-1/2}$. Let $\mathfrak{h}(\mathbf{t},\mathbf{x};\mathfrak{h}_0)$ denote the scaling-limit process appearing in the section, and let the scaling limit be the convergence in finite-dimensional distributions stated in equation. Long-caterpillar threshold conjecture. There exists $b'_0>0$ such that the scaling limit holds if and only if $b'\geq b'_0$. The conjecture predicts a sharp separation threshold: sufficiently distant caterpillars do not alter the stated scaling limit, whereas closer ones do. The source motivates this from the limiting kernel and exploratory computations, but does not determine the value of $b'_0$.