The minimum-degree conjecture for colour-biased loose Hamilton cycles

Let $H$ be a red/blue coloured $3$-graph on $n$ vertices, and let $\delta_1(H)$ denote its minimum vertex degree. A loose Hamilton cycle is a cyclic ordering of the vertices in which every consecutive block of three vertices beginning at positions congruent to $1$ modulo $2$ forms an edge; it has $n/2$ edges. The minimum-degree conjecture for colour-biased loose Hamilton cycles. For every $\alpha>0$, there exist $\delta,n_0>0$ such that, whenever $n\geq n_0$ and

$$\delta_1(H)\geq \left(\frac{7}{16}+\alpha\right)\binom{n}{2},$$

$H$ contains a loose Hamilton cycle with at least $(\frac{1}{4}+\delta)n$ edges of the same colour. This is proposed as the colour-biased analogue of the known asymptotically sharp minimum vertex-degree threshold $\frac{7}{16}\binom{n}{2}$ for loose Hamilton cycles in $3$-graphs; the source gives no resolution.