The nuclear norm $\|\cdot\|_*$ is a norm and hence a convex function. On the other hand, the set $$S:=\{X\in\mathbb R^{m\times n}\colon X_{ij}\ge0\text{ and }\sum_{j=0}^n X_{ij}=1\ \forall i,j\}$$ is convex and compact. So, the maximum of the nuclear norm $\|\cdot\|_*$ on the set $S$ is attained at an extreme point of $S$, which is clearly a matrix $X\in S$ such that one entry of each row of $X$ is $1$ and the other entries are $0$. (You have to allow the non-strict inequalities $X_{ij}\ge0$ for the maximum to be attained.)

The nuclear norm $\|\cdot\|_*$ is a norm and hence a convex function. On the other hand, the set $$S:=\{X\in\mathbb R^{m\times n}\colon X_{ij}\ge0\text{ and }\sum_{j=1}^n X_{ij}=1\ \forall i,j\}$$ is convex and compact. So, the maximum of the nuclear norm $\|\cdot\|_*$ on the set $S$ is attained at an extreme point of $S$, which is clearly a matrix $X\in S$ such that one entry of each row of $X$ is $1$ and the other entries are $0$. (You have to allow the non-strict inequalities $X_{ij}\ge0$ for the maximum to be attained.)