In this paper from 2010 https://cmuc.karlin.mff.cuni.cz/pdf/cmuc1001/arhangav.pdf Arhangelskii asks if there is a quotient of a second countable locally compact Hausdorff space which is Frechet-Urysohn and Tychonoff but not first countable at any point (Problems 5.14 and 5.15).
Is this problem still open?
