What is largest class of functions $C$ such that we know $\#P$ in not contained in $C$-generated $TC^0$? In [1] it is stated that 


  "It remains an open question as to whether every function in $\#P$ has $TC^0$ circuits (although it is at least known that not all $\#P$ functions have DLogTime-uniform $TC^0$ circuits)."


$TC^0$ circuits generated by DLogTime functions does not contain $\#P$.
We don't know if $TC^0$ circuits generated by arbitrary functions does not contain $\#P$.

Is there anything known about the cases in between these two?
E.g. is it known if $TC^0$ circuits generated by $L$ does not contain $\#P$?


[1] Agarwal, Allender, and Datta, "On $TC^0$, $AC^0$, and Arithmetic Circuits"
This is an (interesting) open problem, as far as I know. Rahul Santhanam and I explicitly mention the problem of proving Permanent is not in LOGSPACE-uniform TC0 in our CCC'13 paper (On Medium-Uniformity and Circuit Lower Bounds).