Locally uniform convergence conjecture for almost periodic stochastic processes

Let $f\in B_2\cap C(\mathbb{R},\mathbb{C})$ satisfy the linear independence condition and assume either the condition ensuring continuous sample paths of $\mathcal{M}_f$, or the two stated summatory conditions. Then the locally uniform convergence conjecture.

$$(f(V_{-L,L}+t))_{t\in\mathbb{R}}\Longrightarrow(\mathcal{M}_f(t))_{t\in\mathbb{R}},\quad L\to+\infty,$$

on $C(\mathbb{R},\mathbb{C})$ endowed with the topology of locally uniform convergence. The conjecture asks whether the finite-dimensional convergence can be strengthened to functional convergence when both processes have almost surely continuous sample paths; the source gives no resolution.