The equivalence of Host–Kra, Weyl, true and algebraic complexity

Let $t\in\mathbb{N}_+$ and let $\vec{P}\in\mathbb{R}[x,y]^{t+1}$ be an integral polynomial progression. For each $0\leqslant i\leqslant t$, let $\mathcal{HK}_i(\vec{P})$, $\mathcal{W}_i(\vec{P})$, $\mathcal{T}_i(\vec{P})$, and $\mathcal{A}_i(\vec{P})$ denote respectively the Host–Kra, Weyl, true, and algebraic complexities at $i$. Complexity-equivalence conjecture. Then

$$\mathcal{HK}_i(\vec{P})=\mathcal{W}_i(\vec{P})=\mathcal{T}_i(\vec{P})=\mathcal{A}_i(\vec{P})\leqslant t-1.$$

The four notions measure, respectively, the characteristic-factor, higher-order Fourier-analytic, finitary Gowers-norm, and algebraic complexity of a polynomial progression. The conjecture would give a purely algebraic criterion for determining the smallest Host–Kra factor or the smallest Gowers norm controlling the progression.