(LF)-spaces with more-than-separable quotients

The separable quotient problem remains open for Banach spaces. The recent solution for barrelled spaces counters Saxon-Narayanaswami's solution for (LF)-spaces. W.J. Robertson proposed properly separable quotients. We fully answer her quarter-century-old question, proving (LF)-spaces admit properly separable quotients with rare exception. Corollary: Every (LF)-space except φ admits properly separable quotients if and only if Banach's classic separable quotient problem has a positive solution.