Weak hypercyclicity criterion for bilateral weighted shifts

Let $1\leq p<\infty$, and let $T$ be the bilateral weighted shift on $\ell^p(\mathbb{Z})$ defined by

$$T(e_j)=w_je_{j-1}$$

for a bounded positive weight sequence $(w_j)_{j\in\mathbb{Z}}$. Weak hypercyclicity conjecture. The operator $T$ is weakly hypercyclic if and only if there exists a sequence $(n_k)$ of integers such that

$$\sup\{w_jw_{j-1}\cdots w_{j-n_k+1}:k\geq1,j\in\mathbb{Z}\}<\infty,$$

and, for every $j\in\mathbb{Z}$,

$$w_jw_{j+1}\cdots w_{j+n_k}\longrightarrow\infty\quad\text{as }k\to\infty.$$

The proposed criterion was motivated by the weak hypercyclicity criterion, but it is false: no bilateral weighted shift satisfies both conditions (1) and (2) of the conjecture.