I would like to know whether the following conjecture is true, possibly with additional assumptions such as unconditionality.
Conjecture 1. Suppose $(x_i)_{i=1}^\infty$ is a normalized basis for a Banach space $X$ which is dominated by the unit vectors in $\ell_p$. Then $X$ contains a seminormalized basic sequence dominated by the canonical unit vectors $((e_k^{(n)})_{k=1}^n)_{n=1}^\infty$ in $(\oplus\ell_2^n)_p$.
Discussion.
The conjecture is trivial for $p\geq 2$, so if necessary we may assume $1\leq p<2$.
By Dvoretsky's Theorem we can find $(u_k^{(n)})_{k=1}^n\subset X$ which is 2-equivalent to $(e_k^{(n)})_{k=1}^n$ for each $n\in\mathbb{N}$, and such that $\text{supp }u_k^{(m)}<\text{supp }u_j^{(n)}$ whenever $m<n$.
Can we prove that $((u_k^{(n)})_{k=1}^n)_{n=1}^\infty$ is dominated by $((e_k^{(n)})_{k=1}^n)_{n=1}^\infty$?