The low dropout means the driver can output AT MOST 8V, but as it is regulated, it will give any voltage blow 8V that will create the desired current. In other words, it'll still work with 6.5V input.
For 803T diodes, there is a very large diversity between seemingly identical diodes.
To get an idea, here's a 'maximum current' list for 16 803T diodes I could get my hands on:
76
120
132
135
140
144
156
161
167
170
170
170
180
190
190
198
So if this list would (and of course it won't) represent the power distribution of all 803Ts, then if you drive them by 120mA, there is a 6-7% change you're unlucky.
[edit] all diodes needed slightly over 5V to operate. Less than 5.5V for sure.