= Solution
Write $U(x,\tau)=T(x,t_f+\tau)$ after removal, with $\tau\geq0$. Neglecting heat loss to air means thermal insulation, rather than an end to internal <thermal conduction>. The evolution and conditions are
$$
\boxed{U_\tau=\kappa U_{xx},\quad0<x<L,\quad U_x(0,\tau)=U_x(L,\tau)=0,\quad U(x,0)=T(x,t_f)}.
$$
Here $T(x,t_f)$ is the nonuniform bath-cooled profile derived in part (c). At the switching instant it need not have zero endpoint derivatives: the sudden replacement of <Dirichlet boundary conditions> by <Neumann boundary conditions> can produce another initial-boundary corner incompatibility. The insulated <heat equation> nevertheless recovers the given profile and satisfies its new boundary conditions for $\tau>0$.
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