If you have numerical evidence in support of the conjecture, the journal of Experimental Mathematics seems to fit the bill:
Experimental Mathematics publishes original papers featuring formal results inspired by experimentation, conjectures suggested by experiments, and data supporting significant hypotheses.
Note that we do value proofs: experimentally inspired results that can be proved are more desirable than conjectural ones. However, we do publish significant conjectures or explorations in the hope of inspiring other, perhaps better-equipped researchers to carry on the investigation. The objective of Experimental Mathematics is to play a role in the discovery of formal proofs, not to displace them.
Several publications in that journal have gotten quite some traction, like: