Backward propagation conjecture for cubically hyponormal matrix-valued weighted shifts

Let $W_{\mathcal A}$ be a cubically hyponormal matrix-valued weighted shift, and let $x\in\mathcal H$. Suppose that

$$A_kx=A_{k+1}x$$

for some $k\geq 1$. Backward propagation conjecture. Then

$$A_nx=A_1x$$

for every $n\geq 0$. This conjecture concerns propagation of equality between consecutive weights backward through a cubically hyponormal matrix-valued weighted shift; its resolution is not determined by the supplied text.