(I posted this answer initially at the SSE-original question)
Working with othogonal eigenvector matrices M (created as random rotation matrices) a sequence of experiments suggested, that "varimin"-rotation (which is just minimizing the same criterion which "varimax" maximizes) can run into local extrema and miss the global minimum, and does this more often as the matrix-size is increased.
I used 4x4 up to 22x22 matrices M and produced 1000 and more randomly created examples of the same size, trying to rotate them to "minimal variance of the squared matrix-entries".
For sizes n=4k the "varimin" rotation found always the "Hadamard"-matrix versions.
For sizes n=4k+2 that rotation found sometimes the "Weighing-matrices" with entries from $(1,0,-1)$ (when appropriately rescaled). However, when the dimension went to 18x18 I got that ideal solutions only in roughly 2 percent of the random examples; and for 22x22 matrices that was even less frequent.
Here I checked the improvement of the rotation-criteria for each single iteration and stopped, when the improvement became just marginal but the distance from the optimal versions was still far away.For sizes n x n with n=2k+1 the rotations found sometimes versions which reminded of Hadamard- resp. Weighing- matrices, however had 4,5 or more different entries. For instance for n=13 I got in 80 percent of all examples rotated matrices with only 4 siginificantly different entries in the form $(1,a,-a,-1)$ but for n=11 that result occured only in 1 to 2 percent of examples.
Well, I considered then, that possibly such examples, which did not rotate to the "simple" structure of Hadamard or Weighing matrix having $(1,-1)$ or $(1,0,-1)$ or at least having few-categories-solutions like $(1,a,-a,-1)$ , just are incompatible with suchoptimal solutions. But for the matrix-sizes ( n=4k+2 ) for which I got both types of results, also the bad results could be transformed to an ideal solution by just (the correct) rotation.
The last observation let's me thus conjecture, that indeed "varimin" is not guaranteed to find the solution with the smallest variance in the squared matrix-entries, and the same should then be valid for the "varimax"-rotation in their intention to find the maximal variance.
This is in contrast to one remark of S. Mulaik in his book on "foundations of factoranalysis", which I cite directly from a comment in SSE:
Mulaik (2010, p. 304) said, for Quartimax algorithm, "No proof has ever been offered that such an iterative procedure will always converge, but with empirical structure matrices it always has". This is the only remark on this problem (whether varimax converges to the global optimum) which I've come across so far; funny ...
Well, this experiments were done on orthogonal random-matrices, which are also eigenvector matrices for correlation-matrices. Rotation in factor-/component-analysis is usually done on *loadings*, which are rescaled entries of the eigenvectors - but there are also software packets which indeed rotate on the eigenvectors.