DeBiasio's conjecture on the minimum total degree for powers of Hamilton cycles

Let $D$ be a digraph, and let $\delta(D)$ denote its minimum total degree, namely the minimum number of arcs incident with a vertex. For $k\in\mathbb{N}$ and sufficiently large $n\in\mathbb{N}$, write

$$n=(k+3)q+r,$$

where $q$ is a nonnegative integer and $0\leq r\leq k+2$. The $k$th power of a Hamilton cycle is obtained by including the directed arcs corresponding to consecutive forward distances at most $k$ along the cycle.

DeBiasio's conjecture. Every $n$-vertex digraph $D$ satisfying

$$\delta(D)\geq \begin{cases} 2\left\lceil\left(1-\frac{1}{k+3}\right)n\right\rceil-3, & \text{if }r=k+2,\\ 2\left\lceil\left(1-\frac{1}{k+3}\right)n\right\rceil-2, & \text{if }r\in\{k,k+1\},\\ 2\left\lceil\left(1-\frac{1}{k+3}\right)n\right\rceil-1, & \text{otherwise}, \end{cases}$$

contains the $k$th power of a Hamilton cycle.

The conjecture proposes the minimum total-degree threshold for powers of Hamilton cycles in digraphs. The source presents it as a conjecture but gives no resolution evidence, so its status is left open.