Existence of arbitrarily many metastable critical points of $g_{\xi}$

For integers $m,n_1,\dots,n_m$ and parameters $\nu_{1}^{(1)},\dots,\nu_{n_1}^{(1)},\nu_{1}^{(2)},\dots,\nu_{n_2}^{(2)},\dots,\nu_{n_m}^{(m)}$, consider the functions $g_{\xi}$ arising in the bootstrap-percolation model. A critical point $q_i$ is called $(\nu_{1}^{(i)},\dots,\nu_{n_i}^{(i)})$-metastable at $x_{1}^{(i)},\dots,x_{n_i}^{(i)}$ when it has the corresponding metastability behavior at those points. Metastability conjecture. There exists a function $g_{\xi}$, parameters $\{q_i\}_{i=1}^{m}$, and points $\{x_j^{(i)}\}_{1\leq i\leq m,\,1\leq j\leq n_i}$ such that each $q_i$ is a critical point which is $(\nu_{1}^{(i)},\dots,\nu_{n_i}^{(i)})$-metastable at $x_{1}^{(i)},\dots,x_{n_i}^{(i)}$. The claim formalizes the expectation that suitable offspring distributions can produce arbitrarily many decreasing local maxima of $g_{\xi}$ and hence multiple phase transitions; the supplied text does not establish it or indicate a resolution status.