The monotonicity conjecture for wired percolation on the Kertész line

For $q\in[1,\infty)$, let $p_c(q,h)$ be the critical edge parameter and let $\phi^1_{p,q,h,\mathbb{Z}^d}$ denote the wired random-cluster measure. The wired-percolation monotonicity conjecture. The map

$$h\longmapsto\phi^1_{p_c(q,h),q,h,\mathbb{Z}^d}[0\leftrightarrow\infty]$$

is decreasing. This concerns the infinite-cluster probability along the Kertész line; the source presents it as an open problem for all $q\geq1$.