Exact anti-Ramsey number of vertex-disjoint triangles

Let $kC_3$ denote the union of $k$ vertex-disjoint copies of the triangle $C_3$. For a positive integer $n$, let $ar(n,kC_3)$ be the maximum number of colors in an edge-coloring of $K_n$ containing no rainbow copy of $kC_3$. Conjecture on vertex-disjoint triangles.

$$ar(n,kC_3)=\max\left\{\binom{3k-1}{2}+n-3k+1,\left\lfloor\frac{(n-k+2)^2}{4}\right\rfloor+(k-2)(n-k+2)+\binom{k-2}{2}+1\right\}$$

for all $n\geq 3k$. The formula is known at $n=3k$ and for all $n\geq 2k^2-k+2$, while the intermediate range is the subject of the conjecture.