Nested nodal loops of biharmonic functions

Given any $n\in\mathbb{N}$, we construct a real-valued biharmonic polynomial on $\mathbb{R}^2$ whose zero set contains a nest of $n$ smooth, disjoint topological loops, meaning that the $k$-th loop lies inside the domain bounded by the $(k+1)$-st loop for $k=1,\ldots,n-1$. The case $n=2$, i.e., the existence of two nested loops, is related to the failure of the Boggio-Hadamard conjecture from the early 1900s.