Conjecture on differentiability of the ISE density

Let $f_{\text{\sc ise}}$ be the density of the integrated super-Brownian excursion (ISE), whose existence and Hölder continuity are established in the paper. ISE density differentiability conjecture. Almost surely, $f_{\text{\sc ise}}$ has a continuous derivative but does not have a second derivative. The preceding theorem gives Hölder continuity of every order less than $1$ and an almost-everywhere derivative in $L^p$ for every $2\le p<\infty$; continuity of the derivative and failure of twice differentiability remain conjectural.