Yes, I think there's no harm in initiating occasional warm gestures as long as you can do so without expectation. Maybe sending a pic of the kids if they are doing something nice - or passing on something you think might be of interest to her.
With things like this, only do them if you can truly shrug at the response and not be annoyed, or devastated or thrilled - just notice it and move forward.
If you do send a little something and you get a terse reply - you can always go back to minimal, pleasant, business and reassess at some future point.
Hope your week is going better than you thought it might my friend :-)
T 13 M 7 Me 48 H 46 SS 15 BD 7.14 PA D final 5.16 (H filed)
We receive & we lose, and must try to achieve gratitude & embrace with whole hearts whatever of life that remains after the losses - Dubus