Polynomial Fourier decay for nonlinear self-conformal measures on analytic curves

Let $u$ be a self-conformal measure on $\mathbb{C}$ that is not conjugate to a linear system and is supported on an analytic curve that is not a line. Polynomial Fourier decay conjecture. The measure $u$ has polynomial Fourier decay. This extends the known result for curve-reducible self-conformal measures conjugate to linear systems; the corresponding claim for measures not conjugate to linear remains open and may require quantitative versions of existing arguments.