A famous conjecture of Looijenga states that the moduli space of curves $M_{g,n}$ is the union of $g- \delta_{0,n}+ \delta_{0,g}$ open affine subsets, where $g,n$ are non-negative integers satisfying $2g-2+n>0$, and $\delta$ is the Kronecker delta.

I know of proofs of this conjecture in the case $(g,0)$ for $2 \leq g \leq 5$ (Fontanari-Looijenga and Fontanari-Pascolutti), and in the case $(0,n)$ for all $n$'s.

Is this conjecture true for $M_{1,n}$ ? Namely, is it known if $M_{1,n}$ is affine?

(Are there other cases when the conjecture is known?)