Consider a smooth $k$-dimensional foliation of the unit ball $B$ of $\mathbb{R}^n$, all of whose leaves are diffeomorphic to $k$-disks.

**Question**: Is there a leaf whose $k$-volume is at least $\omega_k$?

Here $\omega_k$ denotes the volume of the unit ball of $\mathbb{R}^k$.

*Particular cases*. If $k=1$, the leaf through the origin (in this case a curve) has the desired property. If $k=n-1$, the leaf which divides the ball into two regions of equal volume has the desired property, by a relative isoperimetric inequality.

*A possible argument*. Replace each leaf $F$ by a $k$-submanifold which minimizes the $k$-volume among those with boundary $F\cap \partial B$. Among the minimal submanifolds obtained in this way, consider one which passes through the origin: its $k$-volume is at least $\omega_k$ by the monotonicity formula, and the $k$-volume of the leaf with the same boundary is even larger.

I know how to make this argument rigorous when the foliation is close enough to the foliation by parallel affine subspaces (but a reference or a more elementary proof also for this perturbative case would be very useful). In general, besides for possible singularities of the minimizers (which should not disturb), the problem I see is how to guarantee that at least one of them passes through the origin.