Prime-dimension existence conjecture for binary Steiner structures

Let $n$ be a prime satisfying

$$n \equiv 1 \pmod{6},\qquad n\geq 13.$$

A coset complete 3-dimensional subspace is a 3-dimensional subspace of $\mathbb{F}_2^n$ used as a generator for the construction described in the paper. Prime-dimension existence conjecture. There exists a Steiner structure $\mathbb{S}_2[2,3,n]$ generated by a set of $\frac{2^n-2}{42n}$ pairwise disjoint coset complete 3-dimensional subspaces. The constructed example $\mathbb{S}_2[2,3,13]$ provides supporting evidence, but the asserted existence for all indicated primes remains open.