The Zeno product formula conjecture for admissible functions

Let $H$ be the nonnegative self-adjoint operator and $P$ the orthogonal projection from the paper, with ${\mathfrak h}=\operatorname{Ran}P$, and let $K$ be the operator governing the limiting Zeno dynamics. For an admissible function $\phi$ and $f\in{\mathfrak h}$, consider the iterated products $\left(\phi(tH/n)\right)^nf$.

Zeno product formula conjecture. If assumption in Theorem II.6a is dropped, then for every $f\in{\mathfrak h}$, $T>0$, and arbitrary admissible function $\phi$,

$$\lim_{n\to\infty}\int_0^T\left\|\left(\phi(tH/n)\right)^nf-e^{-itK}f\right\|^2\,dt=0.$$

The conjecture proposes that at least this integrated strong convergence survives without assumption; the existing proofs for $\phi(x)=e^{-ix}$ use analytic properties of the exponential, whereas general admissible functions require a different proof.