The eventual chromatic spectrum conjecture for Kirkman triple systems

Let a Kirkman triple system of order $v$ be a resolvable Steiner triple system on $v$ points, and let its chromatic number be the minimum number of colours in a colouring in which every triple receives at least two colours. The admissible orders satisfy $v\equiv 3\pmod{6}$.

Eventual chromatic spectrum conjecture. For every integer $\delta\geqslant 3$, there exists an integer $v_0$ such that whenever $v\geqslant v_0$ and $v\equiv 3\pmod{6}$, there is a $\delta$-chromatic Kirkman triple system of order $v$.

This conjecture is the Kirkman-triple-system analogue of the corresponding result known for Steiner triple systems. It would establish the eventual existence of every chromatic number at every admissible order, but remains open.