*The volume of a Riemannian metric on the projective plane is $2\pi$ and length of every non-contractible loop is greater than $\pi - \epsilon$ for some small, positive number $\epsilon$. Is this metric close to the canonical metric?*


The question is somewhat vague on purpose. I'm mostly interested in the best constant 
for a bilipschitz equivalence in terms of $\epsilon$, but I also wonder whether for some sufficiently small $\epsilon$ one can conclude that the curvature is close to 1. 


Stability of inequalities is a well-trodden research topic in convex geometry and I was wondering what was known about this in systolic geometry. 



Stability of Pu's isosystolic inequality
There is no Lipschitz or even Gromov-Hausdorff stability - just consider a round metric with long hairy tails of small area.


One can hope for stability with respect to intrinsic flat distance in the sense of Sormani-Wenger or some similar metric. This distance is basically Federer's flat distance between isometric images is $L^\infty$ (just like the Gromov-Hausdorff distance is the Hausdorff distance in $L^\infty$). The stability in this sense probably amounts to uniqueness of the equality case in the class of integral current spaces arising as limits of projective planes. 


