Suppose we are given an $L^2(\mathcal{D})$-valued Brownian motion $W_t$ defined by $$W_t:=\sum_{k=1}^{\infty}\sqrt{\sigma_k}W_t^k\phi_k(x),$$ where $\mathcal{D}$ is bounded domain in $\mathbb{R}^d$, $\{\phi_k(x)\}$ forms the complete orthogonal basis of $L^2(\mathcal{D})$, $\{W_t^k\}_{k\in \mathbb{N}^{+}}$ are mutually independent one-dimensional standard BM, and $\sigma_k$ satisfies $$\sum_{k=1}^{\infty}\sigma_k<\infty.$$

Is it possible that

$$\lim_{k\to \infty}\sup_{t\geq 0}(-\Lambda t+\sqrt{\sigma_k}W^k_t)=0,\ \mathbb{P}-a.s.,$$

where $\Lambda$ is an given positive number?