The following counterexample is due to Leon Simon. Take $\mathbb{R}^2$ with coordinate label $(x,y)$. Define a current $T$ supported in $x$-axis $\cup$ $y$-axis of the shape $\lrcorner+\ulcorner.$ To be precise, $\lrcorner$ trace the negative $x$-axis, then the non-negative $y$-axis. $\ulcorner$ traces the negative $y$-axis and then the non-negative $x$-axis. By construction, $T$ is a cycle with no boundaries and can be realized as the boundary of two quadrants.

Let $T_a$ be the unique line parallel to the $x$-axis and passing through $(0,a).$ Then $T_a+T$ is stationary for all $a\not=0$ both as varifolds and currents. However, $T_0+T$ equals two times the nonnegative $x$-axis plus the $y$-axis, and is not stationary.

The following counterexample is due to Leon Simon. Take $\mathbb{R}^2$ with coordinate label $(x,y)$. Define a current $T$ supported in $x$-axis $\cup$ $y$-axis of the shape $-\lrcorner+\ulcorner.$ To be precise, $\lrcorner$ trace the negative $x$-axis, then the non-negative $y$-axis, and the minus sign means reversing orientation. $\ulcorner$ traces the negative $y$-axis and then the non-negative $x$-axis. By construction, $T$ is a cycle with no boundaries and can be realized as the boundary of two quadrants suitably oriented.

Let $T_a$ be the unique line parallel to the $x$-axis and passing through $(0,a).$ Then $T_a+T$ is stationary for all $a\not=0$ both as varifolds and currents. However, $T_0+T$ equals $\llcorner+\ulcorner$ and is not stationary.