Neurons in our brains die and get replaced all the time and yet, our experience remains fairly unchanged. I don't see why replacing neurons with their synthetic counterparts would work differently.
There still is some concern regarding the speed of replacement, However. There is much difference between killing and regenerating neurons gradually over many years and replacing them all instantly. Even if we conducted such experiment, there would be no way of telling whether the person's "steam of consciousness" have been severed or not, as the result of an unsuccessful consciousness transfer would be a perfect clone of the original brain.