The minimal filling conjecture for simple Riemannian manifolds

Let $N$ be a closed $(n-1)$-dimensional manifold and let $f:N\times N\to\mathbb R$ be a nonnegative function. Define the filling volume by

$$\operatorname{FillVol}(N,f)=\inf\{\operatorname{Vol}(M):\partial M=N,\ bd_M\ge f\}.$$

A compact Riemannian manifold $M$ is a minimal filling if

$$\operatorname{Vol}(M)=\operatorname{FillVol}(\partial M,bd_M).$$

Minimal filling conjecture. Every simple manifold is a minimal filling. This is the main conjecture of the lecture and concerns whether simple metrics minimize volume among all fillings with boundary distances at least those of the given metric. The source also explains that limits of simple metrics motivate extensions beyond strict simplicity, and that the conjecture would imply the hemisphere case of the circle filling problem.