Exponential-field-size conjecture for lower triangular Toeplitz superregular matrices

Let $\gamma\geq 2$, and let $\mathbb{F}$ be a finite field. A lower triangular Toeplitz superregular matrix is a lower triangular Toeplitz matrix whose relevant minors are all nonzero. Exponential-field-size conjecture. There exists a lower triangular Toeplitz superregular matrix of order $\gamma\times\gamma$ over $\mathbb{F}$ whenever

$$|\mathbb{F}|\geq 2^{\frac{2\gamma}{3}}.$$

This conjecture proposes an exponential upper bound on the field size needed for matrices of arbitrary order; the paper presents it alongside the preceding conjecture based on its computed examples, without resolving it.