Each of us carries a representation of our reality in our heads and it is a totally constructed reality (sound , pictures, memories etc. are all electrical impules, nothing in our brain is playing a little media centre with real speakers and an lcd screen) and that includes our constructed relationships and perceptions of each other. When my wife sees me she does not have an actual me living inside her noggin but a set of neural pathways and electro-chemical reactions describing her view of me. Just as a painting is not the original, neither is our concept of reality.
A conceptual God absolutely exists in the mind of each believer. Most of us here at one stage had a mentally 'real' god in our heads. This god talked to us (via feelings, intuitions or for the select few via actual voices or visions) and we conversed back in various types of prayer and devotion. This inner god absolutely exists but it is a lone god with a beliver of one. The outer god (i.e. a real god existing independently of the imagination who exists independent of human thought) can never be this god of the imagination any more than a painting is a real vase of sunflowers. The proof of my existence is my material presence and its material relationship with the physical world. There are many conceptual versions of me (my mother has one, each of my kids and so on) but only I am real. Each conceptual version of me is different (my wife sees me as a hot , to-die-for, lover with buns of steel while my kids see me as fat, goofy guy who dresses like a scarecrow.) The god in each believer's head is different. No two believers have the same concept of God and the subjective descriptions of their god is next to useless in ascertaining whether a real person called Zeus, with magical powers and some awesome Greek real estate, actually lives.
Proof of me is not found in memories of those who know me. It is found in my material impact on the world. To prove that little green aliens exist we would need to have physical contact. Physical contact is recordable (cameras, audio recordings etc.) Even telepathy would be recordable (we might need to get some new tools to do it) - if it was actually happenning. Thus we must not pretend that god is found in subjective , anecdotal- my prayers were answered, I know love, my god shows me visions type scenarios - but if there is any god(s) they will be found in material ways. The god of the old testament did tangible real things that can be checked (global flood, no death before Adam, creation rather than evolution etc) and since these physical interactions have been proven not to have happened we can safely reject the OT god. The NT god did some recordable things and we can see if they happened (foundational stories such as the census in the nativity and the supposed rulers alive during that story can be checked - guess what didn't happen.) If we limit ourselves to physical interaction stories we can see time and time again that certain biblical events simply did not/do not happen (no mass opening of the graves with lots of resurrections - no return of a flying messiah within a 70 year generation - no miraculous healings shown as signs of those who follow 'him' and so on)
We are left with imagined gods who absolutely do not interact with reality and therefore are simply personal (powerful and life changing!) mental constructs. Mine radically chanaged my behaviour and lifestyle - this response of mine to a mental image does not prove my God and for any believer who is saying 'There is the proof of God' you must realise that my mental god caused me to preach mormonism door to door and perform masonic style rituals in a temple. JWs mental gods cause them to endanger life due to blood transfusions . Muslim gods encourage a small minority to die for the cause.
The whole debate rests upon each person's willingness to question themselves. This is not a momentary epiphany but an ongoing , learnt, habit. To provide proof of your God you must have evidence of physical interaction otherwise you MUST concede your image of god is constructed entirely by your imagination since nothing external to you has occurred.