Boundary regularity conjecture for Hawkes-process resolvent functions

Let $A>0$, let $\beta>1$, and suppose that the functions $\boldsymbol{h}^0$ belong to $C^\beta([0,A])$. For $(l,j,k)\in[K]^3$, $r\in\{1,\ldots,\lfloor\beta\rfloor\}$, $i\in\{0,\ldots,r-1\}$, and $c\geq 0$, define

$$p^{[2]}_{r,k}(x,s,l,j):=\mathbb{E}^{(x,s,l,j)}_0\left[\frac{1}{\lambda_A^k(f_k^0)^r}\right],$$

and

$$p^{![2]}_{r,i,c,k}(x,s,l,j):=\mathbb{E}^{(x,s,l,j)}_0\left[\frac{1}{\lambda_A^k(f_k^0,!(s,j))^{r-i}\bigl(\lambda_A^k(f_k^0,!(s,j))+c\bigr)^{i+1}}\right].$$

Here $\|\cdot\|_{C^\beta}$ denotes the usual Hölder norm on $[0,A]$.

Boundary regularity conjecture. For every $x\in[0,A]$, there exists a version of $s\mapsto p^{![2]}_{r,i,c,k}(x,s,l,j)$ that is continuous at $0$ and at $A$. Moreover, there exist versions of $x\mapsto p^{![2]}_{r,i,c,k}(x,A,l,j)$ and $x\mapsto p^{![2]}_{r,i,c,k}(x,0,l,j)$ belonging to $C^\beta([0,A])$, and

$$\int_0^A\left\|p^{[2]}_{r,k}(\cdot,s,l,j)\right\|_{C^\beta}\,ds<+\infty.$$

These regularity properties are conjectured uniformly over the displayed choices of indices and parameter $c$; they provide the boundary and Hölder regularity needed for semiparametric analysis of multivariate Hawkes processes. The source does not give evidence that this conjecture has been resolved.