The uniform Lagrangian spectral-width conjecture

Let $N$ be a closed manifold, let $g$ be a Riemannian metric on $N$, and let $D_1T^*N=\{(q,p)\mid |p|_g\leq 1\}$. Let $\mathfrak L(T^*N)$ denote the relevant class of closed exact Lagrangians, and let $\gamma(L)$ denote the Lagrangian spectral invariant used in the source. Uniform spectral-width conjecture. There exists a constant $C_N(g)$ such that, for every Lagrangian $L\in\mathfrak L(T^*N)$ contained in $D_1T^*N$,

$$\gamma(L)\leq C_N(g).$$

The claim asks for a uniform bound on the spectral invariant for all such Lagrangians. The supplied text does not identify a resolution or provide an attribution, so its current status should be checked.