Chromatic-number conjecture for derived STS of SQS(n), n congruent to 2 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. For an STS $S'$, let $\chi(S')$ denote the minimum number of partial parallel classes needed to partition its blocks.

Chromatic-number conjecture for derived STS. If

$$20 \le n \equiv 2 \bmod 6,$$

then there is an SQS$(n)$ such that all its derived STS have chromatic number $n/2$, equivalently, each can be partitioned into $n/2$ partial parallel classes. Equivalently,

$$q'_0(3,4,n)=n/2+1.$$

The conjecture is a design-theoretic existence statement intended to determine the alphabet-size parameter $q'_0(3,4,n)$. The supplied source does not report a general proof or refutation.