Resolvable-derived-STS conjecture for SQS(n), n congruent to 4 modulo 6

Let an SQS$(n)$ be a Steiner quadruple system on $n$ points, and let a derived STS be the Steiner triple system obtained by fixing a point of the SQS and deleting that point from every block containing it. A Steiner triple system is resolvable when its blocks can be partitioned into parallel classes.

Resolvable-derived-STS conjecture. If

$$n \equiv 4 \bmod 6,$$

then there is an SQS$(n)$ such that all its derived STS are resolvable.

This conjecture is used to obtain $q'_0(3,4,n)=n/2$. The source reports it as known for $n=10$, for several families admitting 2-resolvable SQSs, and for sufficiently large $n$ with $n\equiv4\bmod12$, but not in general.