A consistent σ-compact sequential space which is not hereditarily weakly Whyburn

We prove that under a=c (in particular, under Martin's Axiom) there exists a regular σ-compact sequential space which is not hereditarily weakly Whyburn. This gives a consistent solution to a question, first formulated by V.V. Tkachuk and I.V. Yashenko, and then raised again by F. Obersnel.