We will use the notation in [1].
A sequence $(x_n)$ in $X$ is called weakly $p$-summable ($p\ge 1$) if $(x^*(x_n))\in \ell_p$ for each $x^*\in X^*$. Equivalently, a sequence $(x_n)$ in $X$ is weakly $p$-summable if there is a $C>0$ such that $$w_p((x_n))=\sup_k \{ \| \sum_{n=1}^k b_n x_n\|: b=(b_n)\in B_{\ell_{p^*}} \} \le C<\infty. $$
A sequence $(x_n)$ in $X$ is called absolutely $p$-summable ($p\ge 1$) if $$ s_p((x_n))= \left( \sum_{n=1}^\infty \|x_n\|^p \right)^{1/p}<\infty.$$
A sequence $(x_n)$ in $X$ is called strongly $p$-summable ($p\ge 1$) if $$ \sigma_p((x_n))= \sup \{| \sum_{n=1}^\infty f_n(x_n) |: w_{p^*}((f_n))\le 1, (f_n)\in X^* \} <\infty.$$
We will denote by $\ell_p[X]$, $\ell_p\{X\}$, and $\ell_p<X>$ respectively the spaces of weakly $p$-summable, absolutely $p$-summable and strongly $p$-summable sequences in $X$, endowed with their natural topologies induced by the norms $w_p$, $s_p$, and $\sigma_p$, respectively.
The symbols $\pi$ and $\epsilon$ denote the projective and injective norms on the space $\ell_p \otimes X$. The symbol $\Delta_p$ denotes the norm induced by $s_p$ on $\ell_p \otimes X$; the topology induced by $s_p$ is termed the \emph{natural} topology. We will denote by $\ell_p\otimes_\pi X$, and $\ell_p\otimes_{\Delta_p} X=\ell_p\{X\}$ the completion of $\ell_p \otimes X$ with respect to $\pi$, and $\Delta_p$, respectively.
The space $\ell_p\otimes_\pi X$ also admits a representation as a vector sequence space: it is the closed subspace of the space $\ell_p<X>$ formed by those sequences which are limits of their finite sections; this can be deduced from [2], where it is proved that the norm $\sigma_p$ induces $\pi$ on $\ell_p \otimes X$.
Lemma ([1]) Let $1\le p<\infty$. Let $X$ and $Y$ be Banach spaces. Let $A$ be a subset of $\ell_p\otimes_\pi X$ (resp. $\ell_p\otimes_{\Delta_p} X$). The following are equivalent:
(i) For each operator $T:\ell_p\otimes_\pi X\to Y$ (resp. $T:\ell_p\otimes_{\Delta_p} X\to Y$) with representing sequence $(T_k)$,
$\sum_{k=1}^n T_k p_k\to T$ uniformly on $A$.
(ii) $ \lim\limits_{n\to \infty} \sup_{x\in A} \pi [(x_k)_{k\ge n}]=0 $ (resp. $\lim\limits_{n\to \infty} \sup_{x\in A} \Delta_p [(x_k)_{k\ge n}]=0 $).
Let $1\le p<\infty $. A subset $A$ of $\ell_p\otimes_\pi X$ (or $\ell_p\otimes_{\Delta_p} X$) is called almost compact if it satisfies either of the equivalent conditions of the previous Lemma.
Proposition 2 in [1] shows the following:
Let $p<\infty$. A subset $A$ of $\ell_p \otimes_\pi X$ (resp. $\ell_p \otimes_{\Delta_p} X$) is relatively compact if and only if it is bounded, almost compact, and for each $k$, $p_k(A)$ is relatively compact in $X$.
Question. Let $p<\infty$. Let $A$ be a bounded subset of $\ell_p \otimes_\pi X$ such that $A$ is almost compact, and for each $k$, $p_k(A)$ is relatively weakly compact in $X$. Does it follow that $A$ is relatively weakly compact? Is there a characterization of relatively weakly compact sets in $\ell_p \otimes_\pi X$?
[1] J. M. F. Castillo and J. A. L'opez Molina, Dunford-Pettis like properties of projective and natural tensor product spaces, Rev. Mat. Univ. Complut. Madrid, 6 (1993), 233--240.
[2] J. S. Cohen, Absolutely $p$-summing, $p$-nuclear and their conjugates, Math. Ann. 201 (1973), 177-200.
