Strong stability conjecture under double commutation on the synthesis kernel

Let $H$ be a Hilbert space, let $T_1,T_2\in B(H)$ commute, and suppose that $\\{T_1^iT_2^j\varphi\\}_{i,j\geq 0}$ is an overcomplete frame for $H$. Let $R_1$ and $R_2$ be the associated operators, assumed to doubly commute on the kernel of the frame's synthesis operator. Strong stability conjecture. For every $f\in H$,

$$T_1^iT_2^j f\longrightarrow 0\qquad\text{as }i,j\to\infty.$$

This asserts strong stability of the forward iterates of the commuting operator pair under the stated frame and double-commutation hypothesis. The supplied text gives no resolution, so the conjecture remains open.