The density conjecture for the integrated superBrownian excursion

Let $\mu_{\mathrm{ISE}}$ be the integrated superBrownian excursion measure on $\mathbb{R}$, and let $Y(\lambda)$ be the non-negative random variable whose law is specified by the local limit theorem. Write $\operatorname{Leb}$ for Lebesgue measure on $\mathbb{R}$. The density conjecture. There exists a random continuous process $f_{\mathrm{ISE}}(\lambda)$, defined for $\lambda\in\mathbb{R}$, such that

$$\mu_{\mathrm{ISE}}=f_{\mathrm{ISE}}\operatorname{Leb},$$

and

$$f_{\mathrm{ISE}}(\lambda)\stackrel{\mathrm{d}}{=}\frac{1}{\sqrt{2}}Y\left(\frac{|\lambda|}{\sqrt{2}}\right).$$

The conjecture asserts absolute continuity and continuity of the ISE density, with its one-point distributions determined by the local limit law for embedded trees. The existence of such a continuous density was not established in the source, so the conjecture remains open.