Existence of strongly MDS convolutional codes with maximum distance profile

A convolutional code with parameters $(n,k,\delta)$ has length $n$, dimension $k$, and degree $\delta$. It is strongly MDS if it attains the strongly MDS distance bound, and it has a maximum distance profile if its column distances attain the corresponding upper bounds.

Existence conjecture. For all $n>k>0$ and for all $\delta\geq 0$ there exists an $(n,k,\delta)$ code over a sufficiently large field which is both strongly MDS and has a maximum distance profile.

The paper proves existence for all parameters with $k=n-1$ and reports computer searches producing such codes for many small parameter values. The conjecture asks for existence for all admissible parameters.