  The tropical Nevanlinna theory is Nevanlinna theory for tropical functions or
maps over the max-plux semiring by using the approach of complex analysis. The
main purpose of this paper is to study the second main theorem with tropical
hypersurfaces into tropical projective spaces and give a defect relation which
can be regarded as a tropical version of the Shiffman's conjecture. On the one
hand, our second main theorem improves and extends the tropical Cartan's second
main theorem due to Korhonen and Tohge [Advances Math. 298(2016), 693-725]. The
growth of tropical holomorphic curve is also improved to
$\limsup_{r\rightarrow\infty}\frac{\log T_{f}(r)}{r}=0$ (rather than just
hyperorder strictly less than one) by obtaining an improvement of tropical
logarithmic derivative lemma. On the other hand, we obtain a new version of
tropical Nevanlinna's second main theorem which is different from the tropical
Nevanlinna's second main theorem obtained by Laine and Tohge [Proc. London
Math. Soc. 102(2011), 883-922]. The new version of the tropical Nevanlinna's
second main theorem implies an interesting defect relation that
$\delta_{f}(a)=0$ holds for a nonconstant tropical meromorphic function $f$
with $\limsup_{r\rightarrow\infty}\frac{\log T_{f}(r)}{r}=0$ and any
$a\in\mathbb{R}$ such that $f\oplus a\not\equiv a.$
