It is not true.

Let upper triangular matrix $T$ be also a differential, i.e. $T^2=0$. Then, acting by upper triangular conjugation one can get a matrix with at most one $1$ in each row and each column, all other elements equal to zero. Moreover, such a matrix (normal form) is unique in the upper triangular conjugacy class. But from the point of view of all conjugations two such a differentials are equivalent if and only if the dimensions of its homologies coincides.

The simplest example through differentials are matrices

001

000

000

and

000

001

000

They are not in the same upper triangular conjugacy class.

Remark: Such differentials and an action of the upper-triangular group naturally arise in a context of Morse theory. A strong Morse function and a generic Riemannian metric on a closed manifold generates Morse complex with an upper triangular Morse differential (upper triangular with respect to the ordering by critical value). A change of a metric reflects as an upper-triangular conjugation of Morse differential. The statement above (on the unicity of normal form in a case of field coefficients) was formulated and proved by Barannikov and attributed to J.Cerf (S. A. Barannikov, “The framed Morse complex and its invariants”, Adv. Soviet Math. 21. (1994), 93–115.).