Optimal universal hypergraph conjecture for partition-injective tree homomorphisms

Let $D\in\mathbb{N}$. An $(n,d,\lambda)$-graph is a graph on $n$ vertices with degree $d$ and second eigenvalue bounded by $\lambda$. Let $T$ be a tree of maximum degree at most $D$, and let

$$V(T)=T_1\cup\cdots\cup T_\ell$$

be a partition with $|T_i|\leq \alpha n$ for every $i\in[\ell]$. Partition-injective tree homomorphism conjecture. For every $D\in\mathbb{N}$, there exist $d\in\mathbb{N}$ and $\alpha>0$ such that, whenever $G$ is an $(n,d,\lambda)$-graph with $\lambda=\Theta(\sqrt d)$, every such tree $T$ admits a homomorphism $\phi\colon T\to G$ whose restriction to each $T_i$ is injective. If true, this would replace the use of the random branching-walk lemma in the proof of the paper's universal-hypergraph theorem and yield the optimal bound $O(n^{r-r/d})$.