Kuca’s algebraic true complexity conjecture for polynomial progressions

Let $P_1,\dots,P_m\in\mathbf{Z}[y]$ be polynomials and let $0\leq i\leq m$. Consider the polynomial progression

$$x,x+P_1(y),\dots,x+P_m(y).$$

An algebraic relation among these entries is an identity

$$Q_0(x)+Q_1(x+P_1(y))+\dots+Q_m(x+P_m(y))=0,$$

where $Q_0,\dots,Q_m\in\mathbf{Z}[z]$. Kuca’s conjecture. The true complexity at $i$ is the smallest natural number $s$ such that, for every such algebraic relation satisfied by the progression, the degree of $Q_i$ is at most $s$. The source presents this as a polynomial-progression analogue of the Gowers–Wolf conjecture and notes that only special cases are known. The conjecture remains open in general.