The partite Nash-Williams conjecture

Let $(V_1,V_2,V_3)$ be a partition of the vertices of a tripartite graph $G$, with $|V_1|=|V_2|=|V_3|=n$. Define

$$\hat{\delta}(G):=\min\{d(v,V_j):1\leq j\leq 3,\ v\in V(G)\setminus V_j\}.$$

Call $G$ partite-$K_3$-divisible when, for every $v\in V(G)\setminus(V_{j_1}\cup V_{j_2})$, one has $d(v,V_{j_1})=d(v,V_{j_2})$ for all $1\leq j_1<j_2\leq 3$.

Partite Nash-Williams conjecture. For sufficiently large $n$, if $G$ is a balanced partite-$K_3$-divisible $3$-partite graph on $3n$ vertices with

$$\hat{\delta}(G)\geq \frac{3}{4}n,$$

then $G$ admits a $K_3$-decomposition.

This is the tripartite analogue of Nash-Williams' conjecture and is equivalent to a high-density completion statement for partial Latin squares. The source presents it as open and notes that only bounds for its fractional relaxation and conditional transfer results are known.