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Most-perfect magic squares

A magic square is said to be most-perfect, if it contains the consecutive numbers 1, 2, …, n² such that

any four adjacent integers forming a 2x2-subsquare sum to
any pair of integers distant n/2 along a diagonal sum to
T=n² + 1
it is a double-even magic square, i.e. of order n=4k.

The following magic square with order 4 and 8 are most-perfect.

1 15 10 8
12 6 3 13
7 9 16 2
14 4 5 11
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	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	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	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