Shear constraint on virtual eigenvalues of ${\mathbb Z}^2$ subshifts

Let $\Xi$ be a minimal, aperiodic and uniquely ergodic ${\mathbb Z}^2$ subshift, and let $\Omega$ be its suspension. Let $\lambda=(\lambda_x,\lambda_y)$ be a virtual eigenvalue of $\Omega$. A shear constraint conjecture asserts:

$$\begin{array}{ll} \text{if }\Xi\text{ admits a horizontal shear,}&\lambda_x\in {\mathbb Z};\\ \text{if }\Xi\text{ admits a vertical shear,}&\lambda_y\in {\mathbb Z};\\ \text{if }\Xi\text{ admits both,}&\lambda\in {\mathbb Z}^2. \end{array}$$

This conjecture extends the established constraint that a minimal ${\mathbb Z}^2$ subshift with shears in both coordinate directions has topological-eigenvalue spectrum precisely ${\mathbb Z}^2$. The source does not provide evidence of a resolution for the corresponding virtual-eigenvalue statement.