Geometric Chevalley–Warning conjecture in the hypersurface form

Let $k$ be an algebraically closed field of characteristic $0$, and let $f$ be a homogeneous polynomial defining a hypersurface of degree $d$ in $\mathbb{P}^n$. A variety is $\mathbb{L}$-rational when its class in the Grothendieck ring $K_0(\operatorname{Var}_k)$ is congruent to $1$ modulo $\mathbb{L}=[\mathbb{A}^1_k]$. Geometric Chevalley–Warning conjecture. Every hypersurface of degree $d<n+1$ in $\mathbb{P}^n$ is $\mathbb{L}$-rational. This is the hypersurface reformulation of the projective geometric Chevalley–Warning conjecture; the paper proves several important cases but leaves the general assertion open.