Geodesic
Example of the matrix of size $6\times6$ with different tropical rank and Kapranov rank
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (2011), pp. 58-61

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We provide an example of a $6$-by-$6$ matrix $A$ such that $\operatorname{rk}_t(A)=4$, $\operatorname{rk}_K(A)=5$. This disproves the conjecture formulated by M. Chan, A. Jensen, and E. Rubei. 
@article{VMUMM_2011_5_a12,
     author = {Ya. Shitov},
     title = {Example of the matrix of size $6\times6$ with different tropical rank and {Kapranov} rank},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {58--61},
     publisher = {mathdoc},
     number = {5},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_2011_5_a12/}
}
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Ya. Shitov. Example of the matrix of size $6\times6$ with different tropical rank and Kapranov rank. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (2011), pp. 58-61. http://geodesic.mathdoc.fr/item/VMUMM_2011_5_a12/