Nonexistence of extended colorings in doubled Steiner triple systems

Let a binary Steiner triple system, denoted $\operatorname{BSTS}(v)$, be a BSTS of order $v$. Suppose that a BSTS of order $2^{h}-1$ is obtained from a BSTS of order $3$ by a sequence of doubling-plus-one constructions.

Nonexistence conjecture. It has no extended colorings for every $h\geq 1$.

Extended colorings are used to distinguish the upper and lower chromatic numbers of binary Steiner triple systems and to study their feasible sets. The source states that this conjecture remains open.