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This is interesting to me too. In my case, I decided for a simple approach: by posting on the arXiv, here is one such paper. I also created a separate route on my own website. Since then the conjectures have received several attention as you can see at these coordinates where I keep updating new papers proving the claims or certain partial progress. That is one possible choice though, you may still opt to find a home for your work in some journals, such as in Experimental Mathematics as Carlo Beenakker pointed out. Perhaps otheres can name the journals they like suggesting.