Rainbow tight Hamilton cycle conjecture at the minimum degree threshold

Let $k$ and $d$ be positive integers, let $\mu>0$, and let $h_d^{k-1}(k,n)$ be the minimum $d$-degree threshold for a Hamilton tight cycle in an $n$-vertex $k$-graph. Let $\textbf{G}=\{G_i\}_{i\in[n]}$ be a $k$-graph system, and let $\delta_d(G_i)$ denote the minimum $d$-degree of $G_i$.

Rainbow minimum-threshold conjecture. There exists $n_0$ such that, for every $n\geq n_0$, if

$$\delta_d(G_i)\geq h_d^{k-1}(k,n)+\mu\binom{n}{d}$$

for every $i\in[n]$, then $\textbf{G}$ admits a rainbow Hamilton cycle.

This conjecture proposes that, up to an additive error, the rainbow threshold agrees with the ordinary tight Hamilton-cycle threshold. The source gives no resolution, so it remains open.