Decomposition conjecture for empirical-process suprema

Let $(X_i)_{i\leq N}$ be independent identically distributed random variables and let ${\cal F}$ be a countable class of measurable functions. Define

$$Z_f:=\frac{1}{\sqrt{N}}\sum_{i\leq N}\bigl(f(X_i)-\mathbb{E}f(X_i)\bigr),\qquad S_N({\cal F}):=\mathbb{E}\sup_{f\in{\cal F}}|Z_f|,$$

and let $d_p(f,g)=\|f-g\|_p$, where the norms are taken with respect to the law of $f(X_i)$. Empirical-process decomposition conjecture. There is a decomposition ${\cal F}\subset {\cal F}_1+{\cal F}_2$ such that

$$\mathbb{E}\sup_{f_1\in{\cal F}_1}\sum_{i\leq N}|f_1(X_i)|\leq \sqrt{N}S_N({\cal F}),$$ $$\gamma_2({\cal F}_2,d_2)\leq LS_N({\cal F}),\qquad \gamma_1({\cal F}_2,d_\infty)\leq L\sqrt{N}S_N({\cal F}).$$

The claim proposes that every empirical-process supremum admits the same pointwise-plus-chaining decomposition as selector processes. The source gives no resolution, so the conjecture remains open.