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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.

1	16	17	32	53	60	37	44
63	50	47	34	11	6	27	22
3	14	19	30	55	58	39	42
61	52	45	36	9	8	25	24
12	5	28	21	64	49	48	33
54	59	38	43	2	15	18	31
10	7	26	23	62	51	46	35
56	57	40	41	4	13	20	29

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

1	16	57	56	17	32	41	40
58	55	2	15	42	39	18	31
8	9	64	49	24	25	48	33
63	50	7	10	47	34	23	26
5	12	61	52	21	28	45	36
62	51	6	11	46	35	22	27
4	13	60	53	20	29	44	37
59	54	3	14	43	38	19	30

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.

1	16	17	32	53	60	37	44
63	50	47	34	11	6	27	22
3	14	19	30	55	58	39	42
61	52	45	36	9	8	25	24
12	5	28	21	64	49	48	33
54	59	38	43	2	15	18	31
10	7	26	23	62	51	46	35
56	57	40	41	4	13	20	29

1	16	57	56	17	32	41	40
58	55	2	15	42	39	18	31
8	9	64	49	24	25	48	33
63	50	7	10	47	34	23	26
5	12	61	52	21	28	45	36
62	51	6	11	46	35	22	27
4	13	60	53	20	29	44	37
59	54	3	14	43	38	19	30

1	16	17	32	53	60	37	44
63	50	47	34	11	6	27	22
3	14	19	30	55	58	39	42
61	52	45	36	9	8	25	24
12	5	28	21	64	49	48	33
54	59	38	43	2	15	18	31
10	7	26	23	62	51	46	35
56	57	40	41	4	13	20	29

1	16	57	56	17	32	41	40
58	55	2	15	42	39	18	31
8	9	64	49	24	25	48	33
63	50	7	10	47	34	23	26
5	12	61	52	21	28	45	36
62	51	6	11	46	35	22	27
4	13	60	53	20	29	44	37
59	54	3	14	43	38	19	30

1	16	17	32	53	60	37	44
63	50	47	34	11	6	27	22
3	14	19	30	55	58	39	42
61	52	45	36	9	8	25	24
12	5	28	21	64	49	48	33
54	59	38	43	2	15	18	31
10	7	26	23	62	51	46	35
56	57	40	41	4	13	20	29

1	16	57	56	17	32	41	40
58	55	2	15	42	39	18	31
8	9	64	49	24	25	48	33
63	50	7	10	47	34	23	26
5	12	61	52	21	28	45	36
62	51	6	11	46	35	22	27
4	13	60	53	20	29	44	37
59	54	3	14	43	38	19	30