Degree-three recovery thresholds for heterogeneous MRA

Let $K\geq 2$ and let $p$ be the signal dimension in heterogeneous MRA. Define

$$\mathcal{U}=p+2+\left\lfloor\frac p2\right\rfloor+\left\lceil\frac{(p-1)(p-2)}6\right\rceil.$$

Heterogeneous MRA recovery conjecture. Generic unique recovery is possible at degree $3$ if $\mathcal{U}>Kp+K-1$, while generic list recovery is possible at degree $3$ precisely if $\mathcal{U}\geq Kp+K-1$. Equivalently, the latter requires $p\geq1$ for $K=2$, $p\geq12$ for $K=3$, $p\geq18$ for $K=4$, and $p\geq6K-5$ for every $K\geq5$. The conjecture was rigorously verified by exact-arithmetic Jacobian and Hessian tests up to $K=15$ and the corresponding critical values of $p$; the general case remains open.