Low-depth arithmetic complexity of the product of $k$ matrices Is anything known about the size of $\Sigma\Pi\Sigma$ (or other constant depth) circuits for the product of $k$ matrices?

Trivial upper bounds (up to small factors) are:


if $k=2$, then there are $\Sigma\Pi\Sigma$ circuits of size
$n^{\omega}$ (where $\omega$ is the matrix multiplication exponent.)
if $k&gt;2$, then  there are $\Sigma\Pi\Sigma$ circuits of size 
$n^{k+1}$.    
if $k&gt;2$, then  there are $O(log(k))$-depth circuits of size 
$n^{\omega}$.


Is it possible to construct circuits of size $n^{\omega}$ for every $k$, for some fixed depth? Is it even possible to beat the trivial construction?
I am not sure about specifically depth-three lower bounds, but there has been a lot of depth-4 (and 5) lower bounds, usually assuming other constraints as well. For instance (and without any claim of exhaustivity):


Hervé Fournier, Nutan Limaye, Guillaume Malod, and Srikanth Srinivasan. Lower Bounds for Depth-4 Formulas Computing Iterated Matrix Multiplication. 
Suryajith Chillara, Partha Mukhopadhyay. Depth-4 Lower Bounds, Determinantal Complexity : A Unified Approach.
Suman K. Bera, Amit Chakrabarti. A Depth-Five Lower Bound for Iterated Matrix Multiplication.