The openness conjecture for discontinuity on the Kertész line

For parameters $q\geq1$ and $h\geq0$, let $\mathfrak{Z}$ denote the relevant pressure function, and consider the phase transition as a function of $p$. The Kertész-line discontinuity openness conjecture. The set of $(q,h)$ such that the map $p\mapsto\mathfrak{Z}$ is not $C^1$ is open. This would express continuity of the corresponding gap in the infinite-volume order parameter; the source notes that the analogous continuity at $h=0$ is known, while the positive-field case remains open.