Uniqueness conjecture for two-parameter presentations of Q36

Let $\mathcal{P}_9(n_1,n_2;m_1,m_2)$ be a presentation of $Q_{36}$, and let $\mathcal{P}_9^{\operatorname{std}}$ denote the standard presentation.

Q36 presentation conjecture. If $\mathcal{P}=\mathcal{P}_9(n_1,n_2;m_1,m_2)$ presents $Q_{36}$, then

$$\mathcal{P}\simeq\mathcal{P}_9^{\operatorname{std}}$$

or

$$\mathcal{P}\simeq\mathcal{P}_9(2;-2).$$

This is a heuristic-computation conjecture about the possible homotopy types of presentations of $Q_{36}$ within the specified family. The supplied text gives no resolution, so the claim remains open.