The random-current Kertész-line monotonicity conjecture

Let $P_h^\emptyset$ denote the random-current measure and let $P_h^{\otimes2}$ denote the sourceless double random-current measure on $\mathbb{Z}^d$. The random-current monotonicity conjecture. For every $d\geq2$, both maps

$$h\longmapsto P_h^\emptyset\left[0\overset{\mathbb{Z}^d}{\leftrightarrow}\infty\right]$$

and

$$h\longmapsto P_h^{\otimes2}\left[0\overset{\mathbb{Z}^d}{\leftrightarrow}\infty\right]$$

are increasing. If true, this would provide the monotonicity needed to establish the existence of a Kertész line for random currents, despite the failure of full monotonicity; the source leaves the conjecture open.