Coordinatewise monotonicity of connectivity on the integer lattice

Let $d\geq 1$ and $p\in(0,1)$. On the graph $\mathbbm{Z}^d$, whose vertices at Euclidean distance $1$ are joined by an edge, use the componentwise order $u\leq v$ defined by $u_i\leq v_i$ for every $i\in\{1,\ldots,d\}$. Coordinatewise monotonicity conjecture. For Bernoulli percolation with parameter $p$,

$$\forall o,u,v\in\mathbbm{Z}^d:\quad o\leq u\leq v\quad\Rightarrow\quad \mathbbm{P}_p(o\leftrightarrow u)\geq\mathbbm{P}_p(o\leftrightarrow v).$$

This folklore conjecture is trivial for $d=1$ but remains open already for $d=2$; a partial result establishes it along coordinate axes when $p$ is sufficiently close to $0$.