Yes at 3" apart, you should be perfectly fine. I was under the impression that the slats you have were 5" apart, in line with what the manufacturer’s minimum requirement indicated. At 3" or less, you should be quite alright.
Just remember to do normal maintenance and rotate the mattress every 3 months for the first year to ensure even wear. Natural foams and fibers have a tendency to compress and develop normal body impressions, and rotating at proper intervals will even those body impressions. This is a good article on mattress maintenance
Maverick