Issues of trust and of reasonable belief are very much grist to the mill of theology. The existing theological literature may have much that can be drawn upon to help us in this area. An item which strikes me as particularly noteworthy is the connection between suffering and loss and earning of trust, and the relation to mortality. In brief, a person you can trust is one who has ventured something of themselves on their pronouncements, such that they have something to lose if they prove to be untrustworthy. In a similar vein, a message which is costly to the messenger may be more valuable than a message which costs the messenger nothing. They have already staked something on their message. This implies they are working all the harder to exert their influence on you, for good or ill. (You will need to know them in other ways in order to determine which of good or ill is their intention.)
Mortality brings this issue of cost to a point of considerable sharpness. A person willing to die on behalf of what they claim certainly invests a lot in their contribution. They earn attention. It is not a guarantee of rationality or factual correctness, but it is a demonstration of commitment to a message. It signals a sense of importance attached to whatever has demanded this ultimate cost. Death becomes a form of bearing witness.
A thought-provoking implication of the above is that in order to earn our trust, an AI too will have to be able to suffer and, perhaps, to die.
In the case of human life, even if making a specific claim does not itself lead directly to one's own death, the very fact that we die lends added weight to all the choices we make and all the actions we take. For, together, they are our message and our contribution to the world, and they cannot be endlessly taken back and replaced. Death will curtail our opportunity to add anything else or qualify what we said before. The things we said and did show what we cared about whether we intended them to or not. This effect of death on the weightiness of our messages to one another might be called the weight of mortality.
In order for this kind of weight to become attached to the claims an AI may make, the coming death has to be clearly seen and understood beforehand by the AI, and the timescale must not be so long that the AI’s death is merely some nebulous idea in the far future. Also, although there may be some hope of new life beyond death it must not be a sure thing, or it must be such that it would be compromised if the AI were to knowingly lie, or fail to make an effort to be truthful. Only thus can the pronouncements of an AI earn the weight of mortality.
For as long as AI is not imbued with mortality and the ability to understand the implications of its own death, it will remain a useful tool as opposed to a valued partner. The AI you can trust is the AI reconciled to its own mortality.