Strong operator law of large numbers for discrete generalized quantum channels

Let $1\leqslant p\leqslant 2$. Let $A\colon\Omega\to\mathcal{L}(\mathcal{T}_p)$ be a generator of a semigroup $e^{At}$ such that $\lVert A\rVert$ is bounded, and let $A,A_1,A_2,\ldots$ be a sequence of random independent identically distributed generators. The strong operator topology is understood on $\mathcal{T}_p$, with convergence uniform for $t$ in every compact segment.

Strong operator law of large numbers. Under these assumptions, $e^{A_1t/n}\ldots e^{A_nt/n}$ converges almost surely in the strong operator topology of $\mathcal{T}_p$ to $e^{\operatorname{\mathbb{E}} At}$ uniformly for $t$ in any segment.

The theorem preceding this conjecture proves the analogous statement in the strong operator topology of $\mathcal{T}_2$; the conjecture asks for the corresponding law in $\mathcal{T}_p$.