# Tokarev's theorem on Banach lattices which are Grothendieck spaces

When browsing the literature, I have found the following theorem of E. Tokarev:

Let $$X$$ be a Banach lattice with weakly sequentially complete dual space. Then for any Banach space $$Y$$, every unconditionally converging operator $$T\colon X\to Y$$ is weakly compact. (In other words, Banach lattices with weakly sequentially duals have Pełczyński's property (V).)

(This is Theorem 1.1 here.)

This theorem, if true, would have a number of fantastic consequences. For instance, a very nice theorem of W. B. Johnson would follow easily:

Suppose that $$X$$ is a Banach space with local unconditional structure. Then either $$X$$ is super-reflexive or $$X$$ contains $$\ell_\infty^n$$'s uniformly or $$X$$ contains $$\ell_1^n$$'s uniformly complemented.

I can produce a rather lengthy list of further applications (some of them, I believe, new and which would be useful to me for other purposes). However, I must admit I don't understand the proof. (For instance, I don't understand why the measures $$\tilde{\mu}_n$$ are well defined, and why we can pass to a convergent subsequence of $$y_n^*$$'s.)

After a while, I decided to apply the so-called psychological argument: if this theorem is so strong, let us look on other results quoting it. Unfortunately, neither google nor mathscinet can find anything. I also tried to contact the author but it seems he doesn't check his mailbox. Hence my question:

Does this theorem have a chance to be true? Or maybe there is an easy counter-example?