Has it been demonstrated that the memory of human brain is just connections between neurons, not "long" molecules such as RNA or proteins?
The idea that memories could be stored as RNA or proteins is an old one. It got a lot of attention decades ago when James McConnell did a number of experiments where he conditioned planarian flatworms to respond to certain stimuli, ground them up, fed them to worms that hadn't gone through the conditioning. He claimed to have observed that the worms fed conditioned worms learned to respond to the stimuli faster. This supported the idea that memories were just RNA or proteins, and thus survived the grinding up process. His results were not readily reproducible, and so the idea fell out of favor.
There have been a number of studies at this point that have been able to look at the firing of individual neurons and shown that there are unique patterns that respond to different stimuli, for example Category-specific visual responses of single neurons in the human medial temporal lobe. So I think it is pretty well established that there are certain connections between neurons that are critical, sometimes even individual neurons.
That doesn't necessarily rule out that there are unique RNA or proteins in those specific neurons, and they are what's actually storing the memory. I'm not a neuroscientist, but I think the biggest argument against such a storage system is simply the speed with which you can recognize something. When you see a face you know you nearly instantly recognize the person, and have a flood of memories about that person. If the memories were somehow stored in RNA or protein structures, I think it would take much longer to somehow "read-out" that data from the molecular structure. The only thing that really works that fast are the action potentials in neurons.
An interesting comparison might be made to the system in our bodies that DOES have molecular-based memory, the immune system. When you get sick with a cold you've never been sick with, your immune system doesn't immediately recognize the antigens that come from that virus. Through the process of somatic hypermutation, your body produces B- and T-cells that have protein receptors that recognize the viruses and the cells infected by the viruses and go and kill them. A small subset of the T and B cells that have the molecular recognition of that cold become 'memory' cells. These memory cells that go into a quiescent phase, waiting around until the next time that cold infects you. But that time they 'remember' it immediately, and you don't get nearly as sick, if at all, because the genes for those recognition proteins are stored in the memory cells. The activation of immunological memory cells takes a long time as compared to remembering a face, or some other normal memory function, so such a molecular system couldn't be responsible.