Oscillation of partition functions under very strong disorder

Let $d \ge 2$, $\mathcal R = \{e_1, \dots, e_d\}$, and let $(X_n)_{n \ge 0}$ be the random walk on $\mathbb Z^d$ from $0$ with independent steps uniform on $\mathcal R$; $E_0$ denotes expectation over the walk. The environment $\omega = (\omega_x)_{x \in \mathbb Z^d}$ is i.i.d. with values in a Borel set $\Gamma \subseteq \mathbb R$, independent of the walk, and $(T_x\omega)_y = \omega_{x+y}$. The potential is local, $V(\omega, z) = V_o(\omega_0, z)$ for a measurable $V_o : \Gamma \times \mathcal R \to \mathbb R$, and may depend on the step $z$. Assume $V \in \mathcal L$: for all $z, z' \in \mathcal R$, $\mathbb E|V(\omega, z)| < \infty$ and almost surely $$\limsup_{\delta \downarrow 0} \limsup_{n \to \infty} \max_{x \in D_1 \cup \dots \cup D_n} \frac1n \sum_{0 \le i \le \delta n} |V(T_{x + iz'}\omega, z)| = 0,$$ where $D_j = \{x \in \mathbb Z_+^d : |x|_1 = j\}$; a finite $p$-th moment of every $V(\cdot, z)$ for some $p > d$ suffices. Let $$Z_n^\omega = E_0 \exp\Bigl\{ \sum_{i=0}^{n-1} V(T_{X_i}\omega, X_{i+1} - X_i) \Bigr\}, \qquad \Lambda_a = \log\Bigl( \frac1d \sum_{z \in \mathcal R} \mathbb E\, e^{V(\omega, z)} \Bigr) \in (-\infty, \infty].$$ The quenched free energy $\Lambda_q = \lim_{n \to \infty} \frac1n \log Z_n^\omega$ exists almost surely and is deterministic, and $\Lambda_q \le \Lambda_a$; very strong disorder means $\Lambda_q < \Lambda_a$. Rassoul-Agha, Seppäläinen and Yilmaz showed that, when $\Lambda_q < \infty$, each of the events $\{\liminf_n e^{-n\Lambda_q} Z_n^\omega = 0\}$ and $\{\limsup_n e^{-n\Lambda_q} Z_n^\omega = \infty\}$ has probability $0$ or $1$, and, using a theorem of Borodin, Corwin and Remenik, that the $\limsup$ is almost surely infinite for the log-gamma directed polymer on $\mathbb Z^2$ with a small enough parameter, which is in very strong disorder.

Problem. Assume $\Lambda_q < \infty$ and very strong disorder, $\Lambda_q < \Lambda_a \le \infty$. Is it always true that, almost surely, $$\liminf_{n \to \infty} e^{-n\Lambda_q} Z_n^\omega = 0 \quad \text{and} \quad \limsup_{n \to \infty} e^{-n\Lambda_q} Z_n^\omega = \infty \, ?$$