in 1979 Hopcroft/ Ullman wrote that L ⊆ P ⊆ NP ⊆ PSpace is known but L ⊊ PSpace is the only proper (& trivial) containment known although all are conjectured to be proper containments, and "where things still stand" ~4 decades later.
since then is there any known connection(s) between L ⊊ P, P ⊊ PSpace and P ⊊ NP? are they all still thought to be independent, or are there any sign(s) of some interdependency?
motivation: this question is partly inspired by the recent Backurs-Indyk results tying SETH to O(n2) edit distance. SETH is exponential time and edit distance is PTime. (& also somewhat the question proving lower bounds by proving upper bounds)