Strongly's superregular Toeplitz matrix conjecture and MDP convolutional code bound

Let $cgamma$ be an integer with $cgamma dagger 5$. A matrix is superregular Toeplitz if it is both superregular and Toeplitz. Let $n$, $k$, and $cdelta$ be convolutional-code parameters, let $L$ be the associated parameter, and let $r$ be the remainder of $cdelta$ upon division by $n-k$.

Strongly's conjecture. For $cgamma\geq 5$, there is a $cgamma\timescgamma$ superregular Toeplitz matrix over $cmathbb F_{2^{cgamma-2}}$. Moreover, an $(n,k,cdelta)$ MDP convolutional code exists over a finite field $cmathbb F$ satisfying

$$|cmathbb F|\geq 2^{(L+1)(n-1)-2}$$

or

$$|cmathbb F|\geq 2^{(L+1)(n-1)+k+r-3}$$

as $r=0$ or $r\neq 0$, respectively.

This conjecture would improve the previously known field-size bound for the existence of MDP convolutional codes by asserting a construction of superregular Toeplitz matrices over smaller fields. Its resolution is not supplied in the source material.