The spectral boundedness conjecture for bounded gamma-support

Let $N$ be a closed Riemannian manifold and let $DT^*N=\{(q,p)\in T^*N\mid |p|_g\leq 1\}$. For an element $L\in\widehat{\mathfrak L}(T^*N)$, its $\gamma$-support is denoted by $\operatorname{\gamma-supp}(L)$. Bounded-support spectral boundedness conjecture. There exists a constant $C_N$ such that every $L\in\widehat{\mathfrak L}(T^*N)$ with

$$\operatorname{\gamma-supp}(L)\subset DT^*N$$

satisfies

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

This is presented as a natural extension of the geometrically bounded implies spectrally bounded conjecture; no proof or resolution is given.