Characterization of critical points of second Neumann eigenfunctions on triangles

Let $T$ be a Euclidean triangle, and let a second Neumann eigenfunction of $T$ mean a Neumann Laplacian eigenfunction corresponding to the second eigenvalue. A triangle is acute if all its angles are less than $\pi/2$, and it is isosceles with apex angle greater than $\pi/3$ if two of its sides are equal and the angle between them exceeds $\pi/3$.

Critical-point conjecture. If $T$ is not an equilateral triangle, then a second Neumann eigenfunction of $T$ has a critical point if and only if $T$ is an acute triangle that is not isosceles with apex angle greater than $\pi/3$.

The preceding results establish the claim for several classes of triangles, including acute, obtuse, right, and isosceles triangles, but the full characterization remains open.