Almgren–Pitts existence conjecture for epsilon-min-max varifolds

Let $X$ be a compact Riemannian manifold of dimension $n$ isometrically embedded in $\mathbb{R}^N$, and let $1\leq k\leq n$. For $\varepsilon>0$, let $[\Pi]$ be a homotopy class of mappings into $C_k(X,G)$, and let $L_{\varepsilon}([\Pi])$ denote the associated epsilon-min-max value. A stationary integral varifold is an element $V\in V_k(X)$ stationary in $X$; being $G$ almost minimizing in a region has its usual Almgren–Pitts meaning. Epsilon-min-max varifold conjecture. There exists a stationary integral varifold $V\in V_k(X)$ such that

$$\lVert V\rVert(\mathbb{R}^N)=L_{\varepsilon}([\Pi]),$$

and, moreover, for every $x\in X$, there exists $r>0$ such that $V$ is $G$ almost minimizing in $X\cap A_N(x,s,r)$ for every $0<s<r$, where $A_N(x,s,r)$ is the annulus in $\mathbb{R}^N$ centered at $x$ with the corresponding radii. This proposes an Almgren–Pitts-type existence and regularity theorem for the discontinuous epsilon-neighborhood volume functional. The author explicitly presents it as a belief for sufficiently small positive $\varepsilon$; no resolution is given in the source.