Quantum law of large numbers in the strong operator topology of ℓp

Let $1\leqslant p\leqslant 2$. Let $A\colon\Omega\to\mathcal{L}(\ell_p)$ be a generator of a semigroup $e^{At}$ such that $\lVert A\rVert_p$ lies in a bounded ball. Let $A_1,A_2,\ldots$ be independent identically distributed generators whose distribution coincides with that of $A$. For $t\geqslant 0$, consider the compositions $e^{A_1t/n}\cdots e^{A_nt/n}$.

Quantum law of large numbers. Under these assumptions, $e^{A_1t/n}\cdots e^{A_nt/n}$ converges in probability in the strong operator topology of $\ell_p$ to $e^{\operatorname{\mathbb{E}}At}$, uniformly for $t$ in every compact segment.

The theorem preceding this conjecture establishes the analogous convergence in the strong operator topology of $\ell_2$. The conjecture asks whether the same law of large numbers holds in the stronger target topology of $\ell_p$.