Consider a sequence $V_N$ of subspaces of $\ell^N_1$ so that $\dim V_N = N- n$ and $n$ is $\mathsf{o}(N)$. Is it true that these spaces are "thick" (unofficial terminology), i.e. are there constants $K_1,K_2>0$ so that for $N$ large enough, $$ \sup_{x \in V, \|x\|_1 \leq K_1} \|x\|_\infty \geq K_2 $$ where $\|x\|_\infty$ is (as usual) the supremum of the coordinates of $x$.

A relaxation (or equivalent formulation?) of the above: is there a sequence $x_i \in V_i \subset \ell_1^i \subset \ell_1\mathbb{N}$ which is convergent in the weak$^*$ topology of $\ell_1 \mathbb{N}$ to something else than $0$?

In the $\ell_2$ case, the answer is "overwhelmingly positive". Let $\lbrace e_i\rbrace_{i=1}^N$ be the "standard" basis of $\ell^N_1$. Consider $P_{V_N}$ to be the orthogonal projection on $V_N$, then $$ N-n = \dim V_N = \sum_{i=1}^N \langle P_{V_N} e_i \mid e_i \rangle $$ So that, in fact, there are at least $N-2n$ coordinates $i$ so that $\langle P_V e_i \mid e_i \rangle \geq \frac{1}{2}$ and $\langle P_{V_N} e_i \mid e_i \rangle = \| P_{V_N} e_i\|_2^2 \leq 1$.