Rainbow tight Hamilton cycle conjecture for Dirac hypergraphs

Let $k\geq 3$ and $\gamma>0$. A colored $k$-graph $(\mathcal{H},\phi)$ has minimum codegree $\delta_{k-1}(\mathcal{H})$, and for each color class $\mathcal{H}_i$ write $\Delta_0(\mathcal{H}_i)$ for its number of edges and $\Delta_{k-1}(\mathcal{H}_i)$ for its maximum codegree. A properly colored tight Hamilton cycle $C_n^{(k)}(k-1)$ is a tight Hamilton cycle whose intersecting edges receive distinct colors.

Rainbow tight Hamilton cycle conjecture. For every $k\geq 3$ and $\gamma>0$ there exist $c>0$ and $n_0>0$ such that if $(\mathcal{H},\phi)$ is an $n$-vertex colored $k$-graph with $n\geq n_0$, $\delta_{k-1}(\mathcal{H})\geq(1/2+\gamma)n$, $\Delta_0(\mathcal{H}_i)\leq cn^{k-1}$ and $\Delta_{k-1}(\mathcal{H}_i)\leq cn$ for every $i\in\mathbf{N}$, then $(\mathcal{H},\phi)$ contains a properly colored tight Hamilton cycle $C_n^{(k)}(k-1)$.

This is a proposed rainbow analogue of the paper's Dirac-type results for properly colored tight Hamilton cycles. The source presents it as a conjecture and does not provide evidence of a resolution.