Conjecture on the refined reconstruction formula for discrete Radon transform data

Let $x_0$ be a generic point, let $H_0$ have well-behaved level sets in the sense that any open interval of length $L\ge L_0$ contains at most $\rho L$ points of any level set $H_0^{-1}(\hat t)$, and assume the hypotheses of the paper's main resolution theorem. Let $K$ be the reconstruction kernel, $f_\epsilon$ the reconstructed function, and $f_\epsilon^{\mathrm{rec}}$ the reconstructed output. For $\check x$ in a compact set, the proposed refined formula is Refined DTB conjecture.

$$f_\epsilon^{\mathrm{rec}}(x_0+\epsilon\check x)=\frac{1}{\epsilon^2}\iint K\left(\frac{(x_0+\epsilon\check x)-y}{\epsilon}\right)f_\epsilon(y)\,\mathrm{d}y+O(\epsilon^{1/2}\ln(1/\epsilon)),\qquad \epsilon\to0,$$

with the big-$O$ term uniform for $\check x$ in any compact set. This conjecture is guided by numerical evidence and the preceding partial lemma; the supplied text does not state that it has been proved or disproved.