The acylindrical PD3-pair splitting conjecture

Let $(G,\{H_r\})$ be an essential white vertex pair in the decomposition under discussion. Suppose that $g\in G$, $s_i\in H_r$, and $s_j\in H_j$ satisfy

$$g^{-1}s_i g=s_j.$$

Acylindrical vertex-pair conjecture. Either $s_i=1$ (and hence $s_j=1$), or $i=j$ and $g^{-1}H_rg=H_r$.

The statement is proposed as the expected acylindricity property of essential white vertex pairs in the $it{PD}^3$ setting. The source gives no evidence of a proof or disproof.