I did the 2 by 2 case for $m$ up to $100.$ For very small entry bound $m,$ a $\pm 1$ is always required in order to get determinant $\pm 1.$ I think the limit of $r_2(m)$ is $0$ as $m$ goes to $\infty.$
I think for $r_3(m)$ you also get limit $0.$ And so on. It should not be difficult switching to $3$ by $3,$ it will just execute even more slowly. 7:16 pm.
jagy@phobeusjunior:~$ ./units
m H_2(m) G_2(m) r_2(m)
1 40 40 1
2 104 104 1
3 232 232 1
4 328 360 0.9111111111111111
5 520 616 0.8441558441558441
6 616 744 0.8279569892473119
7 840 1128 0.7446808510638298
8 968 1384 0.6994219653179191
9 1192 1768 0.6742081447963801
10 1320 2024 0.6521739130434783
11 1608 2664 0.6036036036036037
12 1704 2920 0.5835616438356165
13 1992 3688 0.5401301518438177
14 2152 4072 0.5284872298624754
15 2408 4584 0.525305410122164
16 2568 5096 0.5039246467817896
17 2888 6120 0.4718954248366013
18 2984 6504 0.4587945879458795
19 3336 7656 0.4357366771159875
20 3496 8168 0.4280117531831538
21 3784 8936 0.423455684870188
22 3944 9576 0.4118629908103592
23 4296 10984 0.3911143481427531
24 4424 11496 0.3848295059151009
25 4776 12776 0.3738259236067627
26 4968 13544 0.3668044890726521
27 5256 14696 0.3576483396842678
28 5416 15464 0.3502327987584066
29 5832 17256 0.3379694019471488
30 5928 17768 0.3336334984241333
31 6344 19688 0.3222267370987403
32 6504 20712 0.3140208574739282
33 6792 21992 0.308839578028374
34 7016 23016 0.3048314216197428
35 7400 24552 0.3014011078527207
36 7560 25320 0.2985781990521327
37 7944 27624 0.2875760208514335
38 8104 28776 0.2816235752015568
39 8456 30312 0.2789654262338348
40 8616 31336 0.2749553229512382
41 9096 33896 0.2683502478168516
42 9192 34664 0.2651742441726286
43 9608 37352 0.2572285285928465
44 9832 38632 0.2545040381031269
45 10120 40168 0.2519418442541326
46 10344 41576 0.2487973831056379
47 10760 44520 0.2416891284815813
48 10888 45544 0.2390655190584929
49 11368 48232 0.2356941449659977
50 11560 49512 0.2334787526256261
51 11912 51560 0.2310318076027928
52 12072 53096 0.2273617598312491
53 12488 56424 0.2213242591804906
54 12648 57576 0.2196748645268862
55 13128 60136 0.2183051749368099
56 13352 61672 0.2165001945777663
57 13704 63976 0.2142053269976241
58 13864 65768 0.2108016056440822
59 14344 69480 0.206447898675878
60 14440 70504 0.2048110745489618
61 14920 74344 0.2006886904121382
62 15144 76264 0.1985733766914927
63 15464 78568 0.1968231341003971
64 15752 80616 0.1953954549965267
65 16200 83688 0.1935761399483797
66 16360 84968 0.1925430750400151
67 16776 89192 0.1880886178132568
68 16936 91240 0.1856203419552828
69 17352 94056 0.1844858382240367
70 17512 95592 0.1831952464641393
71 18120 100072 0.1810696298664961
72 18216 101608 0.1792772222659633
73 18696 106216 0.1760186789184304
74 18920 108520 0.1743457427202359
75 19208 111080 0.1729204177169608
76 19496 113384 0.1719466591406195
77 19912 117224 0.1698628267249027
78 20072 118760 0.169013135735938
79 20616 123752 0.1665912470101493
80 20808 125800 0.1654054054054054
81 21224 129256 0.1642012749891688
82 21416 131816 0.1624688960368999
83 21896 137064 0.1597501896924065
84 22056 138600 0.1591341991341991
85 22536 142696 0.1579301452037899
86 22760 145384 0.1565509271996919
87 23112 148968 0.1551474142097632
88 23272 151528 0.1535821762314556
89 23880 157160 0.1519470603206923
90 24040 158696 0.1514845994858093
91 24584 163304 0.1505413217067555
92 24808 166120 0.1493378280760896
93 25096 169960 0.1476582725347141
94 25320 172904 0.1464396428075695
95 25800 177512 0.1453422867186444
96 25960 179560 0.144575629316106
97 26504 185704 0.1427217507431181
98 26728 188392 0.1418743895706824
99 27176 192232 0.1413708435640268
100 27400 194792 0.1406628608977781
jagy@phobeusjunior:~$
