It is conjectured that four mutually unbiased bases in dimension 6 do not exist in quantum information. The conjecture is equivalent to the nonexistence of some three $ 6\times 6 $ 6 × 6 complex Hadamard matrices (CHMs) with Schmidt rank at least 3. We investigate the $ 6\times 6 $ 6 × 6 CHM U of Schmidt rank 3 containing two nonintersecting identical $ 3\times 3 $ 3 × 3 submatrices V, i.e. $ U=\frac {1}{\sqrt 2}\left \begin {smallmatrix} W & V \\ V & X\end {smallmatrix}\right $ U = 1 2 W V V X . We show that such U exists, V, W, X have rank 2 or 3, and they have rank 2 at the same time. We construct the analytical expressions of U when V is, respectively, of rank 2, unitary and normal. We apply our results to the conjecture by showing that U with some normal V is not one of the three $ 6\times 6 $ 6 × 6 CHMs.