Solving block band Toeplitz system

I wish to solve the system $T.X=Y$ where is block band Toeplitz and the unkown. In my case, has a particular form: the blocks are symmetric and quite large ($10^5.10^5$ or even more) and I have around $n$ blocks on the block diagonal of $T$, and around $m$ block lower (and upper) diagonals ( $T$ is band block), such that, for example for $n=5$ and $m=2$:

$T=begin{pmatrix}
T_0 & T_1 & T_2 & 0 & 0
bar{T_1} & T_0 & T_1 & T_2 & 0
bar{T_2} & bar{T_1} & T_0 & T_1 & T_2
0 & bar{T_2} & bar{T_1} & T_0 & T_1
0 & 0 & bar{T_2} & bar{T_1} & T_0
end{pmatrix},$ where the bar means the conjugate.

The LU decomposition works quite well, but I would like to exploit this particular structure: symmetric blocks, block Toeplitz and block band matrix.
However, the size of the blocks is large, and algorithms like block Levinson in are not efficient at all since they fully invert a block $n$ times. They are very efficient for small blocks.

Any idea?