Lang's real analogue of the Tsen–Lang conjecture

Let $X$ be an integral variety over $\mathbb R$ of dimension $i$ such that $X(\mathbb R)=\varnothing$. Recall that a field $K$ is $C_i$ if every degree-$d$ hypersurface in $\mathbb P^N_K$ with $d^i\leq N$ has a $K$-point.

Lang's conjecture. The function field $\mathbb R(X)$ is $C_i$.

This is a real analogue of the Tsen–Lang theorem. Very little is known in general; the case $i=1$, $d=2$ is classical, and Lang proved the assertion for odd degrees $d$. The paper proves the case $i=2$, $d=2$.