  In the Minmax Set Cover Reconfiguration problem, given a set system
$\mathcal{F}$ over a universe and its two covers $\mathcal{C}^\mathsf{start}$
and $\mathcal{C}^\mathsf{goal}$ of size $k$, we wish to transform
$\mathcal{C}^\mathsf{start}$ into $\mathcal{C}^\mathsf{goal}$ by repeatedly
adding or removing a single set of $\mathcal{F}$ while covering the universe in
any intermediate state. Then, the objective is to minimize the maximize size of
any intermediate cover during transformation. We prove that Minmax Set Cover
Reconfiguration and Minmax Dominating Set Reconfiguration are
$\mathsf{PSPACE}$-hard to approximate within a factor of
$2-\frac{1}{\operatorname{polyloglog} N}$, where $N$ is the size of the
universe and the number of vertices in a graph, respectively, improving upon
Ohsaka (SODA 2024) and Karthik C. S. and Manurangsi (2023). This is the first
result that exhibits a sharp threshold for the approximation factor of any
reconfiguration problem because both problems admit a $2$-factor approximation
algorithm as per Ito, Demaine, Harvey, Papadimitriou, Sideri, Uehara, and Uno
(Theor. Comput. Sci., 2011). Our proof is based on a reconfiguration analogue
of the FGLSS reduction from Probabilistically Checkable Reconfiguration Proofs
of Hirahara and Ohsaka (2024). We also prove that for any constant $\varepsilon
\in (0,1)$, Minmax Hypergraph Vertex Cover Reconfiguration on
$\operatorname{poly}(\varepsilon^{-1})$-uniform hypergraphs is
$\mathsf{PSPACE}$-hard to approximate within a factor of $2-\varepsilon$.
