Let me state my question in very loose terms to start, then give some details and restate it in more precise terms at the bottom.
Question. Given a regular minimal cone $\mathbf{C}$, can one set up a well-defined min-max problem between surfaces lying on either side?
Let $\mathbf{C}$ be an $n$-dimensional minimal cone in $\mathbf{R}^{n+1}$. Write $\Sigma = \mathbf{C} \cap \partial B$ for its link, which we assume to be embedded in the sphere. This divides the sphere into two halves, and we write \begin{equation} \partial B \setminus \Sigma = U_- \cup U_+. \end{equation}
Let $\Sigma_{\pm} \subset U_{\pm}$ be two embedded hypersurfaces, in the same homology class, and which lie close to $\Sigma$: \begin{equation} \mathrm{dist}(\Sigma_{\pm},\Sigma) \leq \delta \quad \text{ for some small $\delta > 0$.} \end{equation} Let moreover $M_{\pm}$ be two surfaces in $B$ with $\partial M_{\pm} = \Sigma_{\pm}$, for which we assume there exists an open subset $V \subset B$ so that \begin{equation} \mathrm{spt} \, \mathbf{C} \subset V \quad \text{and} \quad \partial V = M_- \cup M_+. \end{equation}
Consider families of currents $(M_t \mid -1 \leq t \leq 1)$ in the closure of $V$ with $M_{\pm 1} = M_{\pm}.$ We assume that they vary continuously with respect to the flat topology. If necessary, they may be assumed continuous in some finer topology. Perhaps additionally one ought to assume that their boundaries are all homologous: \begin{equation} [\partial M_{t}] = [\partial M_\pm] \quad \text{ for all $-1 \leq t \leq 1$.} \end{equation}
Then define the quantity \begin{equation} W = \inf \max_{-1 \leq t \leq 1} \lVert M_t \rVert, \end{equation} where the infimum is over all families of currents satisfying the hypotheses lined out above.
Question. Is this a well-defined min-max construction, in the sense that $W$ is realised by a free boundary minimal surface in $B$? When is it true $\lVert \mathbf{C} \rVert(B) = W$: always, sometimes, never?
Let me write down a few elementary thoughts, all made under the assumption that (some version of) the problem makes sense.
Variationally one could expect $\mathrm{index} \, T \leq 1$, including for perturbations of the boundary. Therefore $\mathbf{C} \neq T$ if the cone is unstable with respect to $C^1_c(B)$, it seems.
If $\delta > 0$ is too large, say large enough that $\Sigma_{\pm} = \partial B_r(p_\pm) \cap \partial B$ for some small radius $r > 0$ and two points $p_\pm \in \partial B$ is legal, then $W = \omega_n$ and is realised by an equatorial disc.
The hypothesis that $M_t \subset \overline{V}$ is crucial. Otherwise one could shrink $M_{\pm}$ down to small discs, in a way that would make $W = \max \{ \lVert M_{-} \rVert, \lVert M_{+} \rVert \}$ likely. Perhaps additionally $V$ would need to be mean convex?
