Comparison-principle conjecture for regime-switching HJB systems with common convex growth

Let $k\geq 2$, let $q_j=p_j/(p_j-1)$, and let $g_j$ and $\mathcal{A}_g^k$ be as in the generalized convex-growth existence and uniqueness conjecture. Suppose additionally that there is a common convex function $g\geq 0$ and $Q:=\max_j q_j$ such that

$$g_j(x)\leq g(x)\leq C(1+|x|^Q)\qquad\text{for all }j=1,\dots,k.$$

Assume also the strict diagonal dominance condition

$$\delta_j+\alpha_{jj}>0\qquad\text{for all }j.$$

Comparison-principle conjecture. Under these assumptions, the comparison principle of the paper's system comparison theorem extends to solutions in $\mathcal{A}_g^k$. This conjecture addresses the principal missing estimate in the generalized convex-growth theory: common growth control and strict diagonal dominance are expected to make the coupled maximum-principle argument work despite differing component exponents.