The zero-free line conjecture for Miller-basis modular forms

For a modular form in the Miller basis, let its associated Faber polynomial be the polynomial whose zeros correspond to zeros of the modular form on the relevant boundary component. Zero-free line conjecture. The modular forms in the Miller basis have no zeros on the line $\Re(\tau)=0$. Equivalently, the associated Faber polynomials have no zeros in

$$[1728,\infty).$$

This conjecture asserts a global zero-free property for the Miller basis and an equivalent real zero-free interval for the associated Faber polynomials. The supplied text gives no resolution or further evidence beyond this equivalence.