Torsion and ground state maxima: close but not the same

Could the location of the maximum point for a positive
solution of a semilinear Poisson equation on a convex domain be
independent of the form of the nonlinearity? Cima and Derrick
found certain evidence for this surprising conjecture.
We construct counterexamples on the half-disk, by working with
the torsion function and first Dirichlet eigenfunction. On an isosceles right triangle the conjecture fails again. Yet the conjecture has
merit, since the maxima of the torsion function and eigenfunction
are unexpectedly close together. It is an open problem to quantify
this closeness in terms of the domain and the nonlinearity