Doney's smoothness conjecture for scale functions of spectrally negative Lévy processes

Let $X$ be a spectrally negative Lévy process with Gaussian coefficient $\sigma^2$ and Lévy measure $Pi$, and let $W$ be its scale function. Define the tail of the Lévy measure by

$$\overline\Pi(x)=\Pi((-\infty,-x)),\qquad x>0.$$

For $k=0,1,2,\ldots$, the following equivalences should hold:

Doney's conjecture.

1. If $\sigma^2>0$, then

$$W\in C^{k+3}(0,\infty)\Longleftrightarrow \overline\Pi\in C^k(0,\infty).$$

2. If $\sigma=0$ and $\int_{(-1,0)}|x|\,\Pi(\mathrm{d}x)=\infty$, then

$$W\in C^{k+2}(0,\infty)\Longleftrightarrow \overline\Pi\in C^k(0,\infty).$$

3. If $\sigma=0$ and $\int_{(-1,0)}|x|\,\Pi(\mathrm{d}x)<\infty$, then

$$W\in C^{k+1}(0,\infty)\Longleftrightarrow \overline\Pi\in C^k(0,\infty).$$

The conjecture predicts the precise loss of smoothness between the Lévy-measure tail and the scale function in the Gaussian, unbounded-variation, and bounded-variation cases. The surrounding results establish only partial cases under additional analytic assumptions, while the unbounded-variation case with no Gaussian component had not been addressed in the cited work.