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



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> 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)](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-125/issue-1/On-Pe%C5%82-czy%C5%84skis-properties-rm-V-and-rm-Vast/pjm/1102700220.full).)
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(This is Theorem 1.1 [here](https://link.springer.com/article/10.1007/BF00969399#page-1).)


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



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> 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.
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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:



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> Does this theorem have a chance to be true? Or maybe there is an easy counter-example?
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Tokarev's theorem on Banach lattices which are Grothendieck spaces
There is a counterexample in


Figiel, T.; Ghoussoub, N.; Johnson, W. B.
On the structure of nonweakly compact operators on Banach lattices. 
Math. Ann. 257 (1981), no. 3, 317–334.


