space complexity of DFA intersection problem

Question

the DFA-intersection computation problem, given two DFAs specified on the input, compute the intersection DFA, or the decision problem to determine its emptiness, turns out to have wider/ deeper significance in computational complexity theory.[1][2] it is O(n²) time with no better lower bounds known, and tight bounds possibly having major implications on multiple complexity class separations. looking for insight/ refs.

what is the best known space complexity of determining DFA intersection (emptiness)? how does the best known algorithm work?

[1] Deciding emptiness of intersection of regular languages in subquadratic time / cstheory SE
[2] Two DFA intersection emptiness connections to SETH & L vs P / cstheory SE