Subset preservation conjecture for multiple monotone submodular functions

Let $l\geq 1$ be a constant, let $f_1,\dots,f_l$ be monotone submodular functions on a ground set $N$, and let $S\subseteq N$ have size $k$ with $f_i(S)\geq V_i$ for every $i\in[l]$. For an integer $m$ satisfying $s\geq m\geq l$, there should exist a set $X\subseteq S$ of size $m$ such that

$$f_i(X)\geq \frac{m-\Theta(1)}{k}V_i,\quad \forall i\in[l].$$

Subset preservation conjecture. Under these hypotheses, such a set $X$ exists. The claim would provide a simultaneous value guarantee for a constant number of monotone submodular objectives when reducing the cardinality of a feasible set; whether this general statement holds remains open.