Global infinite-oscillation conjecture for the linear multiplicative stochastic heat equation

Let $u$ be the solution of the stochastic partial differential equation considered in the paper, and suppose that

$$\sigma(z)=z\qquad\text{for all }z\in\mathds{R}.$$

Let $f:\mathds{R}^3\to\mathds{R}_+$ be a correlation function satisfying the Dalang condition.

Global infinite-oscillation conjecture. There exist such correlation functions $f$ for which

$$\mathrm{P}\left\{ \mathrm{Osc}_{u(t)}(x)=\infty\text{ for all $(t,x)\in(0,\infty)\times\mathds{R}^3$}\right\}=1.$$

This is a further specialization of the global oscillation claim to the coefficient $\sigma(z)=z$. It seeks simultaneous discontinuity of the second kind throughout space-time, beyond the pointwise results proved earlier in the paper.