Some extended searches (see formula and list of 2008) on 3x3 magic squares of squares
by Lee Morgenstern, January 2013.

A complete enumeration of APs up to d = 2.4 x 10^19 plus a partial enumeration beyond that due to the way the scaling worked resulted in 3809 instances of 3 APs with equal d values and all odd entries.

All but the following were eliminated because either two scale factors had a prime in common or a scale factor was a multiple of an 8k+3 prime.

[d = 71831760]
2171  8749 12181  (12, 23,13)
11633 14393 16703  (77, 92,1)
25889 27241 28529  ( 4,165,1)

[d = 71831760]
6227 10517 13507  ( 5, 28,13)
11633 14393 16703  (77, 92,1)
25889 27241 28529  ( 4,165,1)

[d = 2.75 x 10^15]
4052639   52640389   74334361  ( 6625,2958 1)
90999257  105049757  117430993  (10229, 646,1)
172113455  179937845  187435895  (  986,  21,5x37)

A complete enumeration of 3 primitive APs with equal d values resulted in only 5 instances.

[d = 52492440]
1367  7373 10337  (38, 77,1)
5521  9109 11639  (55, 78,1)
17639 19069 20399  ( 5,138,1)

[d = 8.81 x 10^12]
2987849  4211981  5153089  (2035,266,1)
10522583 10933357 11329247  (3306, 61,1)
11330639 11713109 12083479  (3422, 55,1)

[d = 1.03 x 10^13]
961241 3347261 4635119  (1610, 869,1)
6104543 6895333 7604327  (2622, 143,1)
6568753 7309493 7981783  (2002,1817,1)

[d = 3.58 x 10^13]
1647887  6203957  8617577  (2201,1166,1)
8120041 10085069 11725279  (2438,2035,1)
11258281 12748429 14081761  (3565, 198,1)

[d = 3.31 x 10^15]
10629079  58464869  81995759  ( 7238,2465,1)
60764257  83650813 101501833  ( 9077,1122,1)
93079487 109402717 123588503  (10434, 731,1)

No more were found up to d = 6.4 x 10^22.