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 $k>2$, then there are $\Sigma\Pi\Sigma$ circuits of size $n^{k+1}$.
• if $k>2$, 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?