In this paper, we investigate whether Large Language Models (LLMs) exhibit conspiratorial tendencies, whether they display sociodemographic biases in this domain, and how easily they can be conditioned into adopting conspiratorial perspectives. Conspiracy beliefs play a central role in the spread of misinformation and in shaping distrust toward institutions, making them a critical testbed for evaluating the social fidelity of LLMs. LLMs are increasingly used as proxies for studying human behavior, yet little is known about whether they reproduce higher-order psychological constructs such as a conspiratorial mindset. To bridge this research gap, we administer validated psychometric surveys measuring conspiracy mindset to multiple models under different prompting and conditioning strategies. Our findings reveal that LLMs show partial agreement with elements of conspiracy belief, and conditioning with socio-demographic attributes produces uneven effects, exposing latent demographic biases. Moreover, targeted prompts can easily shift model responses toward conspiratorial directions, underscoring both the susceptibility of LLMs to manipulation and the potential risks of their deployment in sensitive contexts. These results highlight the importance of critically evaluating the psychological dimensions embedded in LLMs, both to advance computational social science and to inform possible mitigation strategies against harmful uses.