Bistability at all processivity levels in the minimally bistable ERK subnetwork

Let $p_k,p_{\ell}\in(0,1)$ be processivity levels. Consider the minimally bistable ERK subnetwork with rate constants

$$(k_1^*, k_3^*, k_{\mathrm{cat}}^*, k_{\mathrm{on}}^*, k_{\mathrm{off}}^*, {\ell}_1^*, {\ell}_3^*, {\ell}_{\mathrm{cat}}^*, {\ell}_{\mathrm{off}}^*, m_2^*, m_3^*, n_1^*, n_3^*)\in\mathbb{R}^{13}_{>0},$$

where

$$p_k=\frac{k_{\mathrm{cat}}^*}{k_{\mathrm{cat}}^*+k_{\mathrm{off}}^*},\qquad p_{\ell}=\frac{{\ell}_{\mathrm{cat}}^*}{{\ell}_{\mathrm{cat}}^*+{\ell}_{\mathrm{off}}^*}.$$

Bistability at all processivity levels. For every choice of $p_k,p_{\ell}\in(0,1)$, there is such a rate-constant vector for which the resulting system admits multiple exponentially stable positive steady states in some compatibility class. Numerical evidence suggests this strengthens the paper's multistationarity theorem from multiple positive steady states to bistability, but the assertion is not proved in general.