Gyárfás's extremal conjecture for monochromatic components in Steiner triple systems

Let $\mathcal{S}_n$ be the family of Steiner triple systems on $n$ points. For a Steiner triple system $S$, let $\alpha^*_3(S)$ denote the maximum size of a 3-independent set, and let $\mathrm{mc}_3(S)$ denote the minimum, over all 3-edge-colorings of $S$, of the order of a largest monochromatic component. Gyárfás's conjecture. For all $n\equiv 1,3 \bmod 6$,

$$\max_{S\in \mathcal{S}_n} \alpha^*_3(S)=\left\lfloor\frac{n}{3}\right\rfloor-1$$

and

$$\min_{S\in \mathcal{S}_n} \mathrm{mc}_3(S)=\left\lceil\frac{2n}{3}\right\rceil+1.$$

The conjecture is best possible in light of the cited lower and upper bounds, and the second assertion is known when $n\equiv 3\pmod {18}$; the general case remains open.