Conjecture on nodal lines of second Neumann eigenfunctions on triangles

Let $D$ be a non-equilateral triangle, and let $u$ be a second Neumann eigenfunction on $D$. Call $D$ superequilateral if it is isosceles with aperture angle greater than $\frac{\pi}{3}$, and subequilateral if its aperture angle is smaller than $\frac{\pi}{3}$. Nodal-line conjecture. The nodal line of $u$ ends at a vertex only when $D$ is superequilateral; for every other non-equilateral triangle, the nodal line connects the two longest sides.

This conjecture would strengthen the known result that the nodal line must start on one side and end on another side or vertex connecting the other two sides. The source reports extensive numerical evidence but does not establish the proposed strengthening.