Does "a religious conspiracy" mean "a conspiracy that is about religion"?

No, in this case it means a conspiracy of religious people. Churchgoers tend to think very poorly of spiritualists and mediums, and would want to discredit them.

Thanks.

Charles Bailey (1870-1947) was an Australian medium known for an ability to produce ‘apports’ of great variety. Investigations provide some support for his genuineness, but he was also often caught in dubious circumstances.

A damaging exposure occurred in Grenoble when Albert de Rochas, a French psychical researcher, visited a market where song birds were sold, to try to identify certain live birds that Bailey had supposedly apported. The seller identified Bailey as the person who had bought them from him earlier.

 https://psi-encyclopedia.spr.ac.uk/articles/charles-bailey