Geometric Chevalley–Warning conjecture for projective zero loci

Let $k$ be an algebraically closed field of characteristic $0$, and let $f_1,\ldots,f_l$ be homogeneous polynomials in $k[x_0,\ldots,x_n]$ satisfying

$$\sum_{i=1}^{l}\deg(f_i)<n+1.$$

Let $Z(f_1,\ldots,f_l)\subseteq\mathbb{P}^n$ be their common zero locus, and let $K_0(\operatorname{Var}_k)$ be the Grothendieck ring of varieties over $k$. Geometric Chevalley–Warning conjecture. One has

$$[Z(f_1,\ldots,f_l)]\equiv 1\pmod{\mathbb{L}}$$

in $K_0(\operatorname{Var}_k)$, where $\mathbb{L}=[\mathbb{A}^1_k]$. The paper explains that this is equivalent to the corresponding hypersurface case and proves it for hyperplane arrangements, quadratic hypersurfaces, singular cubic hypersurfaces, and cubic surfaces in $\mathbb{P}^3$, while the general statement remains open.