Modular S-transformation conjecture for the rank-two Nahm sums

For

$$g(\tau)=\bigl(f_{-3/56,0}(q),f_{1/56,1}(q),f_{9/56,1}(q),f_{-3/56,1}(q),f_{1/56,0}(q),f_{9/56,0}(q)\bigr)^{\mathsf T},$$

where $q=e^{2\pi i\tau}$ and

$$S=\begin{pmatrix}\alpha_3&\alpha_2&\alpha_1\alpha_2&-\alpha_1&-\alpha_3\alpha_1&-\alpha_3&\alpha_2\end{pmatrix},\qquad \alpha_k=\sqrt{\frac27}\sin\frac{k\pi}{7},$$

Modular transformation conjecture.

$$g\left(-\frac1\tau\right)=\begin{pmatrix}S&S\S&-S\end{pmatrix}g\left(\frac\tau2\right).$$

The functions are partial rank-two Nahm sums associated with the matrix displayed in the surrounding text. The formula is proposed from the observed inverse-matrix pattern and numerical evidence; no resolution is given.