 Solution

False, even when the HNN extension also has a finite presentation. Take the base $P=F_m\times F_m$ and its Mihailova subgroup $L$ from Question 4(b). Its word problem for a group is soluble: project a word to each free factor and freely reduce both projections, as in part (iv). The base is finitely presented, but the HNN extensionhas insoluble word problem by the explicit reduction in Question 4(b). Namely, for an input word $w$ in the presentation with insoluble word problem, set $p=(1,w)$; thenThe obstruction is undecidable membership in the associated subgroup, not undecidable equality in the base. Soluble base word problem alone does not let one recognize the pinches needed for a normal form theorem for an HNN extension algorithm.