What is the lower critical dimension (LCD) of the bond-diluted Ising model?

It is known that the lower critical dimension (LCD) $d_l$ of the Ising model is $d_l=1$, that the LCD of the Edwards-Anderson model is $d_l=5/2$ (source) and that the LCD of the random field Ising model is $d_l=2$ (source).

My question is: what is the LCD of the bond-diluted Ising model?

The bond-diluted Ising model is defined by the hamiltonian

$$H = – sum_{langle ij rangle} J_{ij} sigma_i sigma_j$$

where the sum runs over nearest neighbors, $sigma_i=pm 1$ are the usual spin variables and $J_{ij}$ is $1$ with probability $p$ and $0$ with probability $1-p$.

Clarification: I know that there can be no phase transition in $d=1$ because the one-dimensional bond percolation threshold is $p_c(1)=1$ and a phase transition is possible only if $p>p_c(d)$ (see Yvan Velenkis’s answer and following comments), but I was wondering whether more general results including non-integer dimensions are available (see this paper and this paper for examples of discussions where the dimension is treated as a real variable).