  Feigin and Shoikhet conjectured in math/0610410 that successive quotients
$B_m(A_n)$ of the lower central series filtration of a free associative algebra
$A_n$ have polynomial growth. In this paper we give a proof of this conjecture,
using the structure of $W_n$-representation on $B_m(A_n)$ which was defined in
math/0610410 . We also prove that the number of squares in a Young diagram $D$
corresponding to an irreducible $W_n$-module in the Jordan-Holder series of
$B_m(A_n)$ is bounded above by the integer $(m-1)^2+2[(n-2)/2](m-1)$. This
bound combined with MAGMA computations by Rains in math/0610410 allows us to
confirm the $W_n$-module structure of $B_3(A_3)$ conjectured in math/0610410 .
