Cutoff at the symmetric boundary case for the SSEP with reservoirs

Consider the symmetric simple exclusion process with reservoirs on a segment of size $N$, with parameters $p$ and $q$, and let $t^*$ denote the characteristic mixing-time scale. In the regime where $p$ is bounded away from $1$, the source states that the cutoff window is of order $\mathcal{O}(1)$, and when $p$ is also bounded away from $0$, $t^*=\frac{\log N}{2\pi^2}+\mathcal{O}(1)$. Boundary-case cutoff conjecture. The system still exhibits cutoff at time

$$\frac{\log N}{2\pi^2}$$

when $p=1$ and $q=0$. This is the boundary case not covered by the stated sufficient condition, and the conjecture proposes that the same cutoff scale persists there.