Conjecture on the monodromy matrix around the critical point

Let $z_c$ be the critical value, and let $\alpha_c$, $\tau_c$, and $d$ denote the quantities appearing in the conjectured monodromy matrix. The monodromy matrix acts in the indicated five-dimensional basis.

Monodromy-matrix conjecture. The monodromy matrix around $z=z_c$ is

$$\frac{1}{\tau_c^2} \begin{pmatrix} \alpha_c^2 & 0 & -\alpha_c(\tau_c^2-\alpha_c^2)/8 & (\tau_c^2-\alpha_c^2)d & -(\tau_c^2-\alpha_c^2)^2/128 \\ -32d & \tau_c^2 & -8\alpha_c d & 32d^2 & -(\tau_c^2-\alpha_c^2)d \\ -8\alpha_c & 0 & \tau_c^2-2\alpha_c^2 & 8\alpha_c d & -\alpha_c(\tau_c^2-\alpha_c^2)/8 \\ 0 & 0 & 0 & \tau_c^2 & 0 \\ -32 & 0 & -8\alpha_c & 32d & \alpha_c^2 \end{pmatrix}.$$

The formula agrees with six examples from the cited table, which supports the conjecture, but the supplied text gives no proof or general resolution.