After millenia of quest we have identified two $n$ bit integers can be multiplied in $O(n\log n)$ time. Please refer details in https://www.quantamagazine.org/mathematicians-discover-the-perfect-way-to-multiply-20190411/. The problem of placing multiplication in $O(n)$ time is still unsolved.
What evidence do we have that this can/cannot be brought to $O(n)$ time on uniform Turning model?
What evidence do we have that this can/cannot be brought to $O(n)$ complexity in a reasonable non-uniform model?