Most-perfect magic squares are pandiagonal

It can be shown that a magic square which satisfies these three conditions, must also be pandiagonal. One example is the following most-perfect magic square of order n=8.

116173253603744
635047341162722
314193055583942
61524536982524
125282164494833
545938432151831
107262362514635
565740414132029

But this doesn't mean on the other side that all pandiagonal magic squares are also most-perfect. For example, the following square is pandiagonal and even more - any four adjacent integers forming a 2x2-subsquare sum to

Supermagisch-Formel n=8

116575617324140
585521542391831
89644924254833
635071047342326
512615221284536
625161146352227
413605320294437
595431443381930

But when you sum any two integers along a diagonal distant n/2=4, you won't get T=65. So this is a pandiagonal, but not a most-perfect magic square.