Advice on supporting friend with BPD

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I'm sure this will be a familiar story: I met a new friend, we were inseparable from pretty much the day we met, then 3 months later she started treating me badly out of the blue, and when I asked to talk about it, she ditched me.

While we were close she had told me she had BPD, which I hadn't heard of. I read the advice on the Mind website which was quite vague and jargonny, and concluded that I just needed to be a good, supportive friend to her and all would be well.

Since being ditched I've done a lot more reading and realise that this is all quite classic behaviour caused by her BPD.

I'd really like to support her and stay friends, I have a lot of love for her and wish I had made myself more informed earlier so I could have seen this coming and understood what was going on at the time.

I'm unsure at this point (a month after being 'split') whether I should reach out. From what I read online she either hates my guts or misses me desperately but fears rejection.

She texts me about once a week, usually practical things, but last time she got in touch it was a friendly text and I could sense a bit of her old self in it. I always reply but as a rule I haven't been initiating conversation with her as that was the thing that really seemed to piss her off after she 'split' me.

So I'd like some advice, please: should I leave her alone and let her initiate friendship if and when she wants to (and accept this might be never), or reach out at some point with a 'thinking of you, hope you're doing well'-type text?