A three-block higher-rank deformed Hermitian--Yang--Mills correspondence on $\mathbb P^1\times\mathbb P^1$

We study the invariant rank-four deformed Hermitian--Yang--Mills equation associated with the chain $\mathcal{O}(1)\to\mathcal{O}(2)^{\oplus2}\to\mathcal{O}(3)$ over $\mathbb P^1$, with vertical weights $(4,2,0)$ and a fixed product Kähler class and phase. For arbitrary linear arrows $ψ=Bz$ and $ϕ=(